Fractional Powers of the Directional Derivative and a Maxwell–Gegenbauer Multipole Identity

Abstract

We study fractional and complex powers of a fixed directional derivative in ${\mathbb{R}}^d$, defined via a Marchaud-type singular integral representation. Under explicit convergence assumptions, this yields a pointwise nonlocal realization along rays. We then formulate a Ramanujan–Hardy approach to fractional directional differentiation based on analytic interpolation of the directional jet at a point. This construction is local in jet space and is governed by Hardy’s formulation of Ramanujan’s Master Theorem. We emphasize that the resulting Ramanujan–Hardy derivative is defined through a Hardy-admissible interpolant of the directional jet. As an application, we investigate fractional directional derivatives of the Newtonian kernel in dimension $d\ge 3$. After a justified regularization and reduction to a Marchaud-type integral, we obtain a one-dimensional integral representation and a zonal harmonic description of the resulting function. This leads to a fractional Maxwell–Gegenbauer identity for $0<\Re \left(s\right)<1$, expressing the fractional directional derivative of ${\parallel x\parallel }^{2-d}$ in terms of Gegenbauer functions of complex degree. In this way, the classical Maxwell multipole formula appears as the integer-order case of a continuous analytic family. Moreover, the fractional operator preserves the main structural properties of the Newtonian kernel, including homogeneity, rotational invariance, and harmonicity away from the origin. The paper thus connects Mellin analysis, Ramanujan’s Master Theorem, fractional calculus, and harmonic analysis on the sphere, while clarifying the distinction between Marchaud and jet-interpolation constructions of fractional directional operators.

1. Introduction

Ramanujan’s Master Theorem (RMT) provides a classical link between formal power series and Mellin transforms. In its standard form, it asserts that, under suitable analyticity and growth assumptions on a function $\varphi$,

$${\int }_0^{\infty }{t}^{s-1}\left[\varphi \left(0\right)-{\displaystyle \frac{t}{1!}}\varphi \left(1\right)+{\displaystyle \frac{t^2}{2!}}\varphi \left(2\right)-\cdots \right]dt=\Gamma \left(s\right)\varphi (-s),0<\Re \left(s\right)<1,$$

(1)

as recorded in Ramanujan’s Quarterly Reports [1] and later placed on a rigorous analytic footing by Hardy [2,3]. The theorem has since become a standard tool in Mellin analysis, summation theory, and analytic continuation, and has also been revisited from operational and umbral viewpoints [4,5].

A formal connection with fractional calculus arises when one interprets the coefficients $\varphi \left(n\right)$ as derivatives of a function at a fixed point. Formally setting $\varphi \left(n\right)=f^{\left(n\right)}\left(x\right)$ leads to integral expressions that recover classical fractional operators such as the Weyl–Liouville and Marchaud derivatives. These operators play a central role in modern fractional calculus; see, for example, refs. [6,7,8,9,10]. More generally, fractional derivatives provide a flexible framework for modeling nonlocal effects and anomalous transport [11]. However, this connection is only formal in general. For a $C^{\infty }$ function, the Taylor series need not converge beyond a local neighborhood, and even when it does, it need not represent $f(x-t)$ for arbitrary $t\ge 0$. Thus the use of (1) in fractional calculus cannot rely on a global Taylor expansion. Rather, it must be understood through analytic interpolation of the derivative sequence together with Hardy’s Mellin transform formulation of Ramanujan’s theorem [2,3]. In particular, RMT should be viewed as an analytic continuation principle for coefficient sequences, rather than as a summation formula for globally convergent series. In parallel, fractional powers of differential operators admit natural definitions via Fourier multipliers and singular integrals. For a fixed direction $\eta \in {\mathbb{S}}^{d-1}$, one may define fractional directional derivatives through the Fourier multiplier

$${(-i\eta ·\xi )}^s,$$

and, under suitable assumptions, represent them by Marchaud-type singular integrals along rays. These constructions are inherently nonlocal and fit naturally into the framework of anisotropic fractional analysis; see [12,13,14,15,16]. The present paper develops a connection between these two viewpoints. On the one hand, we work with the Marchaud realization of the operator

$${(-\eta ·\nabla )}^s,$$

which provides a pointwise nonlocal extension of integer-order directional derivatives along rays. On the other hand, we introduce a Ramanujan–Hardy framework based on analytic interpolation of the directional jet at a point. This second construction is local in jet space and is governed by Hardy’s formulation of Ramanujan’s Master Theorem. It depends on the choice of a Hardy-admissible interpolant of the directional derivatives, and its relation with the Marchaud operator holds only under additional structural assumptions.

Our main motivation comes from harmonic analysis and potential theory. For $d\ge 3$, the Newtonian kernel admits the classical multipole expansion

$${\displaystyle \frac{1}{{\parallel x-y\parallel }^{d-2}}}=\sum _{n=0}^{\infty }{\displaystyle \frac{d-2}{2n+d-2}}{\displaystyle \frac{{\parallel y\parallel }^n}{{\parallel x\parallel }^{n+d-2}}}{Z}_n\left({\displaystyle \frac{y}{\parallel y\parallel }},{\displaystyle \frac{x}{\parallel x\parallel }}\right),\parallel y\parallel <\parallel x\parallel ,$$

(2)

where $Z_n$ denotes the zonal harmonic of degree n. This expansion is a cornerstone of the theory of spherical harmonics and Gegenbauer polynomials; see [17,18,19,20,21]. Termwise differentiation in a fixed direction leads to Maxwell’s classical identity

$${(-\eta ·\nabla )}^n\left({\parallel x\parallel }^{2-d}\right)=\Gamma (n+1){\displaystyle \frac{C_n^{\left(\lambda \right)}(\xi ·\eta )}{{\parallel x\parallel }^{n+d-2}}},\lambda ={\displaystyle \frac{d-2}{2}}.$$

(3)

This identity may be interpreted as a directional form of the multipole expansion and reflects the role of Gegenbauer polynomials in zonal harmonic analysis. A natural question is whether (3) admits a meaningful extension to non-integer, and more generally complex, orders s. Such an extension requires both a rigorous definition of the fractional operator ${(-\eta ·\nabla )}^s$ and an analytic continuation of the harmonic coefficients.

Our approach combines the Marchaud representation of fractional directional derivatives with Hardy’s formulation of Ramanujan’s Master Theorem. This leads to a fractional extension of Maxwell’s identity, which we call the fractional Maxwell–Gegenbauer identity:

$${(-\eta ·\nabla )}^s\left({\parallel x\parallel }^{2-d}\right)=\Gamma (1+s){\displaystyle \frac{C_s^{\left(\lambda \right)}(\xi ·\eta )}{{\parallel x\parallel }^{d+s-2}}},0<\Re \left(s\right)<1.$$

(4)

Here $C_s^{\left(\lambda \right)}$ denotes the Gegenbauer function of complex degree, which interpolates the classical zonal harmonics. A key feature of the result is that (4) is obtained without relying on a global Taylor expansion. Instead, it follows from a combination of nonlocal integral representations and analytic continuation principles. In particular, under suitable assumptions ensuring compatibility between the Marchaud and Ramanujan–Hardy constructions, both approaches lead to the same analytic function of the order parameter s.

The paper thus connects Mellin analysis, Ramanujan’s Master Theorem, fractional calculus, and harmonic analysis on the sphere. It also clarifies the distinction between nonlocal (Marchaud-type) and jet-interpolation (Ramanujan–Hardy) constructions of fractional directional operators, and provides a framework for extending classical harmonic identities to complex orders.

2. Preliminaries

In this section we fix notation and collect the analytic ingredients used throughout the paper. The first part concerns fractional directional operators, represented by Marchaud-type singular integrals. The second part recalls the analytic setting of Ramanujan’s Master Theorem, which will later be used to construct the Ramanujan–Hardy fractional directional derivative. For background on fractional calculus we refer to [6,7,8,9,10], for operator-theoretic and semigroup viewpoints to [15,16], and for the Fourier-analytic framework to [12,13,14]. Classical references on harmonic analysis and special functions that will be relevant later include [19,21,22,23,24].

2.1. Fractional Marchaud Directional Derivatives

Throughout the paper, ${\mathbb{R}}^d$ denotes the d-dimensional Euclidean space equipped with the standard inner product

$$x·y=\sum _{i=1}^d{x}_i{y}_i,x,y\in {\mathbb{R}}^d,$$

and the norm $\parallel x\parallel ={(x·x)}^{1/2}$. The unit sphere is

$${\mathbb{S}}^{d-1}=\{\eta \in {\mathbb{R}}^d:\parallel \eta \parallel =1\}.$$

For a multi-index $j=(j_1,\dots ,j_d)\in {\mathbb{N}}_0^d$, we write $\left|j\right|={\sum }_{i=1}^d{j}_i$ and

$${\partial }^{j}f={\displaystyle \frac{{\partial }^{\left|j\right|}f}{\partial x_1^{j_1}\cdots \partial x_d^{j_d}}}.$$

The gradient and Laplacian are

$$\nabla f=({\partial }_{x_1}f,\dots ,{\partial }_{x_d}f),\Delta f=\sum _{i=1}^d{\partial }_{x_i}^{2}f.$$

Let $\eta \in {\mathbb{S}}^{d-1}$ be fixed. The first-order directional derivative is

$$(\eta ·\nabla )f\left(x\right)=\underset{t\to 0^{+}}{lim}{\displaystyle \frac{f\left(x\right)-f(x-t\eta )}{t}}=\sum _{i=1}^d{\eta }_i{\partial }_{x_i}f\left(x\right),$$

and for integers $n\in {\mathbb{N}}_0$,

$${(\eta ·\nabla )}^{n}f\left(x\right)=\sum _{\left|j\right|=n}{\displaystyle \frac{n!}{j!}}{\eta }^j{\partial }^{j}f\left(x\right).$$

This operator is local. The fractional extensions considered below are intrinsically nonlocal and should be understood pointwise along rays. The next lemma records the one-dimensional Taylor expansion along a fixed direction. It will be used only as a local tool; no global analyticity along rays is assumed unless explicitly stated.

Lemma 1.

Let $\eta \in {\mathbb{S}}^{d-1}$ and $x\in {\mathbb{R}}^d$. Let $m\in {\mathbb{N}}_0$ and assume that $f\in C^{m+1}\left(U\right)$ on an open set U containing the segment

$$\{x+\theta t\eta :\theta \in [0,1\left]\right\}.$$

Then

$$f(x+t\eta )=\sum _{k=0}^m{\displaystyle \frac{t^k}{k!}}{(\eta ·\nabla )}^{k}f\left(x\right)+{\displaystyle \frac{t^{m+1}}{m!}}{\int }_0^1{(1-\theta )}^m{(\eta ·\nabla )}^{m+1}f(x+\theta t\eta )d\theta .$$

If, in addition, f is real analytic near x, then the full series converges for sufficiently small t.

Proof. 

Apply the one-dimensional Taylor formula to $h\left(s\right)=f(x+s\eta )$ and use the identity $h^{\left(k\right)}\left(s\right)={(\eta ·\nabla )}^{k}f(x+s\eta )$. □

Let $0<\Re \left(s\right)<1$ and fix $\eta \in {\mathbb{S}}^{d-1}$. For a measurable function $f:{\mathbb{R}}^d\to \mathbb{C}$ and a point $x\in {\mathbb{R}}^d$, consider the one-dimensional trace

$$g_x\left(t\right):=f(x+t\eta ),t\ge 0.$$

Whenever the following integral converges absolutely, we define the fractional directional derivative of Marchaud type by

$${(-\eta ·\nabla )}^{s}f\left(x\right):={\displaystyle \frac{s}{\Gamma (1-s)}}{\int }_0^{\infty }{\displaystyle \frac{f\left(x\right)-f(x+t\eta )}{t^{1+s}}}dt.$$

(5)

This definition is pointwise and depends only on the behavior of f along the half-line $\{x+t\eta :t>0\}$. A sufficient condition for the absolute convergence of (5) at a given point x is the existence of constants $C_x>0$, ${\alpha }_x>\Re \left(s\right)$, and ${\beta }_x>0$ such that

(i)

for $0<t<1$,

$$|f\left(x\right)-f(x+t\eta )|\le C_x{t}^{{\alpha }_x},$$

(ii)

for $t\ge 1$,

$$\left|f(x+t\eta )\right|\le C_x{(1+t)}^{-{\beta }_x}.$$

Under these assumptions, the integral converges absolutely by comparison with $t^{{\alpha }_x-1-\Re \left(s\right)}$ near 0 and with $t^{-1-\Re \left(s\right)}$ at infinity. The operator (5) is nonlocal and anisotropic: its value at x depends on the restriction of f to a single ray.

Proposition 1.

Let $0<\Re \left(s\right)<1$ and let $\eta \in {\mathbb{S}}^{d-1}$. Assume that the integral (5) defining ${(-\eta ·\nabla )}^{s}f\left(x\right)$ is absolutely convergent.

(i)

Homogeneity. If f is homogeneous of degree κ, then

$${(-\eta ·\nabla )}^{s}f\left(\rho x\right)={\rho }^{\kappa -s}{(-\eta ·\nabla )}^{s}f\left(x\right),\rho >0.$$

(ii)

Equivariance under rotations fixing$\eta$. If $R\in O\left(d\right)$ satisfies $R\eta =\eta$, then

$${(-\eta ·\nabla )}^s(f\circ R)\left(x\right)=\left({(-\eta ·\nabla )}^{s}f\right)\left(Rx\right).$$

Proof. 

Both statements follow directly from (5). For (i), perform the change of variables $t=\rho \tau$ and use the homogeneity of f. For (ii), note that $R\eta =\eta$ implies

$$(f\circ R)(x+t\eta )=f(Rx+t\eta ),$$

and substitute into (5). □

2.2. Analytic Framework for Ramanujan’s Master Theorem

Ramanujan’s Master Theorem lies at the intersection of power series, Mellin transforms, and analytic continuation. Roughly speaking, it asserts that, under suitable analyticity and growth assumptions, the Mellin transform of a locally defined power series recovers the analytic continuation of its coefficient sequence. This principle goes back to Ramanujan’s Quarterly Reports [1] and was placed on a rigorous foundation by Hardy [2,3,25]. It is also related to earlier observations of Glaisher [26] and to later developments in operational and umbral calculus [4,5]. From a broader perspective, it belongs to the general framework of Mellin analysis and analytic interpolation; see, for instance, [27,28,29]. Its interaction with special functions and orthogonal expansions is also well-documented in [21,22,24,30].

The analytic proof uses two classical tools: residue calculus and Mellin inversion. We record the latter in the required form below.

Theorem 1

(Mellin inversion formula). Let F be analytic in a vertical strip $a<\Re \left(s\right)<b$, and define

$$f\left(x\right)={\displaystyle \frac{1}{2\pi i}}{\int }_{c-i\infty }^{c+i\infty }F\left(s\right)x^{-s}ds,$$

for some $c\in (a,b)$. If the integral converges absolutely and uniformly with respect to c on compact subintervals of $(a,b)$, then

$$F\left(s\right)={\int }_0^{\infty }{x}^{s-1}f\left(x\right)dx,a<\Re \left(s\right)<b.$$

We next recall Hardy’s version of Ramanujan’s Master Theorem. It may be viewed as a mechanism that assigns a Mellin-transform meaning to a power series known initially only near the origin.

Theorem 2

(Hardy [2]). Let ϕ be a single-valued analytic function on the half-plane

$$H\left(\delta \right)=\{z\in \mathbb{C}:\Re z\ge -\delta \},0<\delta <1,$$

and assume that

$$\left|\varphi (v+iw)\right|\le C{(1+|v+iw\left|\right)}^B{e}^{Pv+A\left|w\right|},A<\pi .$$

Then, for every s with $0<\Re \left(s\right)<\delta$,

$${\int }_0^{\infty }{x}^{s-1}\left(\varphi \left(0\right)-x\varphi \left(1\right)+x^2\varphi \left(2\right)-\cdots \right)dx={\displaystyle \frac{\pi }{sin\left(\pi s\right)}}\varphi (-s).$$

This theorem should be interpreted with some care. The coefficients $\varphi \left(n\right)$ first appear as the coefficients of a power series defined near $x=0$. The theorem then identifies the Mellin transform of that local series with the analytic continuation of the interpolant at the non-integer point $-s$. In this sense, Ramanujan’s Master Theorem is not a global Taylor theorem: it is an analytic continuation principle for coefficient sequences.

Hardy’s treatment also includes a shifted version, obtained by subtracting a finite number of terms from the local expansion. This improves the behavior of the Mellin kernel near the origin and moves the strip of convergence.

Theorem 3

(Hardy [2]). Under the assumptions of Theorem 2, let $N\ge 1$. Then, for all s such that

$$-N<\Re \left(s\right)<-N+1,$$

one has

$${\int }_0^{\infty }{x}^{s-1}\left(\sum _{n=N}^{\infty }\varphi \left(n\right){(-x)}^n\right)dx={\displaystyle \frac{\pi }{sin\left(\pi s\right)}}\varphi (-s).$$

The proof is obtained by applying Theorem 2 to the shifted function $z\mapsto \varphi (z+N)$ and then reindexing. Conceptually, subtracting the first N terms removes the leading singular behavior at the origin and permits Mellin continuation into a translated strip. Such shifted formulas are closely related to regularization procedures in fractional calculus and singular integral theory; see [6,8,9,10]. This point of view is the one relevant for the fractional directional construction introduced below. Consider a local series of the form

$$\sum _{n=0}^{\infty }\varphi (n,x)t^n.$$

Under the Hardy growth hypothesis, the coefficients have at most exponential growth, so this series converges at least in a neighborhood of $t=0$. In general, there is no reason for it to converge for large t, nor is there any reason for it to coincide globally with a translated function. What Ramanujan’s Master Theorem provides is a Mellin-transform interpretation of that local expansion. This distinction between local series data and global function values will be important in what follows.

3. Ramanujan–Hardy Fractional Directional Derivative

We now introduce a class of functions for which the sequence of directional derivatives along a fixed direction admits a Hardy-admissible analytic interpolation. This leads to a jet-based notion of fractional directional differentiation inspired by Ramanujan’s Master Theorem and Hardy’s analytic formulation [1,2,3,25]. The construction is local in the sense that it depends only on the full directional jet at a single point. It should not, however, be confused with the Marchaud definition, which is nonlocal along a ray.

Definition 1.

Let $\mathrm{\Omega }\subset {\mathbb{R}}^d$ be open, let $\eta \in {\mathbb{S}}^{d-1}$, and fix $x\in \mathrm{\Omega }$. We say that $f:\mathrm{\Omega }\to \mathbb{C}$ belongs to the Ramanujan–Hardy class at x along η, and write

$$f\in {\mathcal{RH}}_{\eta }\left(x\right),$$

if the following conditions hold:

(RH1)

All iterated directional derivatives in the direction $-\eta$ exist at x:

$${(-\eta ·\nabla )}^{n}f\left(x\right),n\in {\mathbb{N}}_0.$$

(RH2)

There exists a function $\varphi (s,x)$ such that:

(a)

$\varphi (·,x)$ is holomorphic in the half-plane

$$H_{\delta }=\{s\in \mathbb{C}:\Re s\ge -\delta \},0<\delta <1;$$

(b)

there exist constants $C,P>0$, $B\in \mathbb{R}$, and $A<\pi$ such that

$$\left|\varphi (s,x)\right|\le C{(1+|s\left|\right)}^B{e}^{P\Re s+A|\Im s|},s\in H_{\delta };$$

(c)

for every $n\in {\mathbb{N}}_0$,

$$\varphi (n,x)={\displaystyle \frac{1}{n!}}{(-\eta ·\nabla )}^{n}f\left(x\right).$$

This definition is modeled on Hardy’s hypotheses for Ramanujan’s Master Theorem. In particular, the interpolant $\varphi (·,x)$ is not arbitrary: it is required to be holomorphic in a half-plane extending to the left of the imaginary axis and to satisfy a Hardy-type growth estimate there. At this stage, existence of such an interpolant is part of the definition of the class ${\mathcal{RH}}_{\eta }\left(x\right)$.

For $f\in {\mathcal{RH}}_{\eta }\left(x\right)$ and a Hardy-admissible interpolant $\varphi$, we associate the local shift series

$${\mathcal{T}}_{\eta }^{\varphi }f(x;t):=\sum _{n=0}^{\infty }\varphi (n,x){(-t)}^n=\sum _{n=0}^{\infty }{\displaystyle \frac{{(-t)}^n}{n!}}{(-\eta ·\nabla )}^{n}f\left(x\right).$$

By the growth bound in Definition 1, this series converges absolutely for $\left|t\right|<e^{-P}$. It therefore defines a holomorphic function near $t=0$. In general, however, it need not converge for all $t>0$, and it need not coincide globally with the translated function $f(x+t\eta )$. This distinction is essential: the Ramanujan–Hardy framework works with the Mellin-transform interpretation of the local series, not with a global translation identity along the ray.

Proposition 2

(Ramanujan–Hardy master formulas). Let $f\in {\mathcal{RH}}_{\eta }\left(x\right)$, and let $\varphi (s,x)$ be a Hardy-admissible interpolant satisfying

$$\varphi (n,x)={\displaystyle \frac{1}{n!}}{(-\eta ·\nabla )}^{n}f\left(x\right),n\in {\mathbb{N}}_0.$$

Then, for every integer $N\ge 1$ and every s such that

$$-N<\Re \left(s\right)<-N+1,$$

one has

$${\displaystyle \frac{\pi }{sin\left(\pi s\right)}}\varphi (-s,x)={\int }_0^{\infty }{t}^{s-1}\left({\mathcal{T}}_{\eta }^{\varphi }f(x;t)-\sum _{n=0}^{N-1}{\displaystyle \frac{{(-t)}^n}{n!}}{(-\eta ·\nabla )}^{n}f\left(x\right)\right)dt,$$

in the sense of Hardy’s formulation of Ramanujan’s Master Theorem.

In particular, for $0<\Re \left(s\right)<\delta$,

$${(-\eta ·\nabla )}_{\mathrm{RH}}^{-s}f\left(x\right)={\displaystyle \frac{1}{\Gamma \left(s\right)}}{\int }_0^{\infty }{t}^{s-1}{\mathcal{T}}_{\eta }^{\varphi }f(x;t)dt,$$

and

$${(-\eta ·\nabla )}_{\mathrm{RH}}^{s}f\left(x\right)={\displaystyle \frac{1}{\Gamma (-s)}}{\int }_0^{\infty }{t}^{-s-1}\left({\mathcal{T}}_{\eta }^{\varphi }f(x;t)-f\left(x\right)\right)dt.$$

Proof. 

By Definition 1, the function $s\mapsto \varphi (s,x)$ is holomorphic in $H_{\delta }$ and satisfies Hardy’s growth condition there. Hence, Theorems 2 and 3 apply to $\varphi (·,x)$. Since

$${\mathcal{T}}_{\eta }^{\varphi }f(x;t)=\sum _{n=0}^{\infty }\varphi (n,x){(-t)}^n,$$

we have

$${\mathcal{T}}_{\eta }^{\varphi }f(x;t)-\sum _{n=0}^{N-1}\varphi (n,x){(-t)}^n=\sum _{n=N}^{\infty }\varphi (n,x){(-t)}^n.$$

Applying Theorem 3 yields

$${\int }_0^{\infty }{t}^{s-1}\left({\mathcal{T}}_{\eta }^{\varphi }f(x;t)-\sum _{n=0}^{N-1}\varphi (n,x){(-t)}^n\right)dt={\displaystyle \frac{\pi }{sin\left(\pi s\right)}}\varphi (-s,x),-N<\Re \left(s\right)<-N+1.$$

This proves the first identity. For $0<\Re \left(s\right)<\delta$, Theorem 2 gives

$${\int }_0^{\infty }{t}^{s-1}{\mathcal{T}}_{\eta }^{\varphi }f(x;t)dt={\displaystyle \frac{\pi }{sin\left(\pi s\right)}}\varphi (-s,x).$$

Using Euler’s reflection formula

$$\Gamma \left(s\right)\Gamma (1-s)={\displaystyle \frac{\pi }{sin\left(\pi s\right)}},$$

we obtain

$$\Gamma (1-s)\varphi (-s,x)={\displaystyle \frac{1}{\Gamma \left(s\right)}}{\int }_0^{\infty }{t}^{s-1}{\mathcal{T}}_{\eta }^{\varphi }f(x;t)dt.$$

This motivates the definition of ${(-\eta ·\nabla )}_{\mathrm{RH}}^{-s}f\left(x\right)$. Finally, applying the shifted identity with $N=1$ and replacing s by $-s$, we obtain

$${\int }_0^{\infty }{t}^{-s-1}\left({\mathcal{T}}_{\eta }^{\varphi }f(x;t)-f\left(x\right)\right)dt=-{\displaystyle \frac{\pi }{sin\left(\pi s\right)}}\varphi (s,x).$$

Since

$$\Gamma (s+1)\Gamma (-s)=-{\displaystyle \frac{\pi }{sin\left(\pi s\right)}},$$

it follows that

$$\Gamma (s+1)\varphi (s,x)={\displaystyle \frac{1}{\Gamma (-s)}}{\int }_0^{\infty }{t}^{-s-1}\left({\mathcal{T}}_{\eta }^{\varphi }f(x;t)-f\left(x\right)\right)dt.$$

This yields the stated positive-order formula. □

Proposition 3.

Let $f\in {\mathcal{RH}}_{\eta }\left(x\right)$, and suppose that ${\varphi }_1(s,x)$ and ${\varphi }_2(s,x)$ are two Hardy-admissible interpolants satisfying

$${\varphi }_1(n,x)={\varphi }_2(n,x)={\displaystyle \frac{1}{n!}}{(-\eta ·\nabla )}^{n}f\left(x\right),n\in {\mathbb{N}}_0.$$

Then

$${\varphi }_1(s,x)={\varphi }_2(s,x),0<\Re \left(s\right)<1.$$

Consequently, the Ramanujan–Hardy fractional directional derivative of order s is well-defined on the strip $0<\Re \left(s\right)<1$, independently of the choice of interpolant.

Proof. 

Since ${\varphi }_1(n,x)={\varphi }_2(n,x)$ for every $n\in {\mathbb{N}}_0$, the associated local shift series coincide:

$${\mathcal{T}}_{\eta }^{{\varphi }_1}f(x;t)={\mathcal{T}}_{\eta }^{{\varphi }_2}f(x;t)$$

for all $\left|t\right|<r$, where $r>0$ is any common disk of convergence. Applying the first formula of Proposition 2 with $N=1$ to each interpolant, and replacing s by $-s$, we obtain for every s with $0<\Re \left(s\right)<\delta$,

$$\Gamma (s+1){\varphi }_j(s,x)={\displaystyle \frac{1}{\Gamma (-s)}}{\int }_0^{\infty }{t}^{-s-1}({\mathcal{T}}_{\eta }^{{\varphi }_j}f(x;t)-f\left(x\right))dt,j=1,2.$$

The right-hand sides are identical because the local shift series are identical. Hence

$$\Gamma (s+1){\varphi }_1(s,x)=\Gamma (s+1){\varphi }_2(s,x),0<\Re \left(s\right)<\delta .$$

Since $\Gamma (s+1)\ne 0$, it follows that

$${\varphi }_1(s,x)={\varphi }_2(s,x),0<\Re \left(s\right)<\delta .$$

More generally, if one wishes to define the Ramanujan–Hardy derivative on a larger positive strip, the same argument may be iterated using the shifted formula with larger values of N. In the present paper, however, only the range $0<\Re \left(s\right)<1$ is needed, and uniqueness on this strip is sufficient. □

We may therefore define the Ramanujan–Hardy fractional directional derivative unambiguously on its natural strip of use.

Definition 2.

Let $f\in {\mathcal{RH}}_{\eta }\left(x\right)$. For $s\in \mathbb{C}$ with $0<\Re \left(s\right)<1$, define

$${(-\eta ·\nabla )}_{\mathrm{RH}}^{s}f\left(x\right):=\Gamma (s+1)\varphi (s,x),$$

where $\varphi (·,x)$ is any Hardy-admissible interpolant of the directional jet of f at x. By Proposition 3, this value is independent of the choice of ϕ.

The normalization is chosen so that at integer orders,

$${(-\eta ·\nabla )}_{\mathrm{RH}}^{n}f\left(x\right)={(-\eta ·\nabla )}^{n}f\left(x\right),n\in {\mathbb{N}}_0.$$

Thus, the Ramanujan–Hardy construction provides an analytic continuation, in the order parameter, of the sequence of integer directional derivatives.

Remark 1.

The quantity ${(-\eta ·\nabla )}_{\mathrm{RH}}^{s}f\left(x\right)$ depends only on the infinite directional jet

$${\left\{{(-\eta ·\nabla )}^{n}f\left(x\right)\right\}}_{n\ge 0},$$

together with the Hardy-admissible interpolation guaranteed by Definition 1. In this sense it is local in jet space, although it involves derivatives of all orders. This locality is conceptually different from the Marchaud derivative, whose value depends on the behavior of the function along an entire ray.

3.1. Spherical and Solid Harmonics

A twice continuously differentiable function f satisfying $\Delta f=0$ on an open subset of ${\mathbb{R}}^d$ is called harmonic. A homogeneous polynomial $P:{\mathbb{R}}^d\to \mathbb{C}$ of degree n is called a solid harmonic of degree n if $\Delta P=0$. The space of solid harmonics of degree n is denoted by ${\mathcal{S}}_n$ and is finite-dimensional. Classical references for this material include [17,18,19,20,23]. For $x\ne 0$, write $x^{\prime }:=x/\parallel x\parallel \in {\mathbb{S}}^{d-1}$. The restriction of $P\in {\mathcal{S}}_n$ to the unit sphere,

$$Y_n\left(x^{\prime }\right):=P\left(x^{\prime }\right),x^{\prime }\in {\mathbb{S}}^{d-1},$$

is a spherical harmonic of degree n. The space of spherical harmonics of degree n is denoted by ${\mathcal{H}}_n$. Every $P\in {\mathcal{S}}_n$ admits the factorization

$$P\left(x\right)={\parallel x\parallel }^n{Y}_n\left(x^{\prime }\right),x\ne 0,$$

so the spherical harmonic records the angular dependence of the solid harmonic. The dimension of ${\mathcal{H}}_n$ is

$$d_n={\displaystyle \frac{(d+2n-2)(d+n-3)!}{n!(d-2)!}},$$

(6)

and one has the asymptotic law $d_n\sim c_d{n}^{d-2}$ with $c_d=2/(d-2)!$. We regard ${\mathcal{H}}_n$ as a subspace of $L^2\left({\mathbb{S}}^{d-1}\right)$, with inner product

$$\langle f,g\rangle ={\int }_{{\mathbb{S}}^{d-1}}f\left(\omega \right)g\left(\omega \right)d\omega .$$

If ${\left\{Y_{n,k}\right\}}_{k=1}^{d_n}$ is a real-valued orthonormal basis of ${\mathcal{H}}_n$, then

$$\{Y_{n,k}:n\ge 0,1\le k\le d_n\}$$

forms a complete orthonormal basis of $L^2\left({\mathbb{S}}^{d-1}\right)$; see [19,20,23].

Fix ${\omega }_0\in {\mathbb{S}}^{d-1}$. Evaluation at ${\omega }_0$ is a continuous linear functional on ${\mathcal{H}}_n$. By the Riesz representation theorem, there exists a unique $Z_n(·,{\omega }_0)\in {\mathcal{H}}_n$ such that

$$Y\left({\omega }_0\right)=\langle Y,Z_n(·,{\omega }_0)\rangle ,Y\in {\mathcal{H}}_n.$$

The function $Z_n(\omega ,{\omega }_0)$ is the zonal harmonic of degree n with pole ${\omega }_0$. Expanding in the orthonormal basis yields

$$Z_n(\omega ,{\omega }_0)=\sum _{k=1}^{d_n}{Y}_{n,k}\left({\omega }_0\right)Y_{n,k}\left(\omega \right).$$

Zonal harmonics admit the Gegenbauer representation

$$Z_n(\omega ,{\omega }_0)={\displaystyle \frac{2(d-2){\pi }^{d/2}}{(2n+d-2)\Gamma (d/2)}}{C}_n^{(d/2-1)}(\omega ·{\omega }_0),$$

(7)

where $C_n^{\left(\alpha \right)}$ denotes the Gegenbauer polynomial of degree n, see [31] (§15.3.1). These polynomials are orthogonal on $[-1,1]$ with respect to the weight ${(1-t^2)}^{\alpha -1/2}$:

$${\int }_{-1}^1{C}_n^{\left(\alpha \right)}\left(t\right)C_m^{\left(\alpha \right)}\left(t\right){(1-t^2)}^{\alpha -1/2}dt={\displaystyle \frac{{\delta }_{mn}}{h_n^{\left(\alpha \right)}}},$$

with an explicit normalization constant $h_n^{\left(\alpha \right)}$. We refer to [21,22,24,30] for classical properties of Gegenbauer functions and related orthogonal expansions.

Let $k:(-1,1)\to \mathbb{C}$ satisfy ${(1-t^2)}^{(d-3)/2}k\left(t\right)\in L^1(-1,1)$. Then, the Funk–Hecke formula asserts that for every $Y_m\in {\mathcal{H}}_m$,

$${\int }_{{\mathbb{S}}^{d-1}}{Y}_m\left(\sigma \right)k(x^{\prime }·\sigma )d\sigma =\widehat{k}\left(m\right)Y_m\left(x^{\prime }\right),$$

(8)

where

$$\widehat{k}\left(m\right)={\displaystyle \frac{2{\pi }^{(d-1)/2}}{\Gamma \left(\right(d-1)/2)}}{\int }_{-1}^{1}k\left(t\right)P_m\left(t\right){(1-t^2)}^{(d-3)/2}dt.$$

A direct consequence is the spherical averaging identity

$${\int }_{{\mathbb{S}}^{d-1}}k(x^{\prime }·\sigma )d\sigma ={\displaystyle \frac{2{\pi }^{(d-1)/2}}{\Gamma \left(\right(d-1)/2)}}{\int }_{-1}^{1}k\left(t\right){(1-t^2)}^{(d-3)/2}dt.$$

For the classical theory of the Funk–Hecke formula and its role in harmonic analysis, see [17,19,20,23]. The next lemma recalls Maxwell’s classical identity, which expresses the nth directional derivative of the Newtonian kernel in terms of zonal harmonics. Variants and consequences of this identity are classical in potential theory, multipole analysis, and mathematical physics; see, for instance, refs. [17,32,33,34].

Lemma 2.

Let $d\ge 3$ and set $\lambda ={\displaystyle \frac{d-2}{2}}$. For every $n\in {\mathbb{N}}_0$ and all $x,y\in {\mathbb{R}}^d$ with $x\ne 0$, one has

$${(-y·\nabla )}^n\left({\parallel x\parallel }^{2-d}\right)=\Gamma (n+1){\displaystyle \frac{{\parallel y\parallel }^n}{{\parallel x\parallel }^{n+d-2}}}{C}_n^{\left(\lambda \right)}\left({\displaystyle \frac{x}{\parallel x\parallel }}·{\displaystyle \frac{y}{\parallel y\parallel }}\right).$$

(9)

Proof. 

Fix $x\ne 0$ and write

$$\xi ={\displaystyle \frac{x}{\parallel x\parallel }}.$$

Let

$$u\left(x\right):={\parallel x\parallel }^{2-d}.$$

For $\left|t\right|<\parallel x\parallel$, the function $t\mapsto u(x-t\eta )$ is analytic, and its Taylor expansion at $t=0$ is

$$u(x-t\eta )=\sum _{n=0}^{\infty }{\displaystyle \frac{{(-1)}^n}{n!}}{(\eta ·\nabla )}^{n}u\left(x\right)t^n,\eta \in {\mathbb{S}}^{d-1}.$$

(10)

On the other hand, using

$${\parallel x-t\eta \parallel }^2={\parallel x\parallel }^2-2tx·\eta +t^2={\parallel x\parallel }^2\left(1-2{\displaystyle \frac{t}{\parallel x\parallel }}(\xi ·\eta )+{\displaystyle \frac{t^2}{{\parallel x\parallel }^2}}\right),$$

we obtain

$$u(x-t\eta )={\parallel x\parallel }^{2-d}{\left(1-2{\displaystyle \frac{t}{\parallel x\parallel }}(\xi ·\eta )+{\displaystyle \frac{t^2}{{\parallel x\parallel }^2}}\right)}^{-\lambda }.$$

By the Gegenbauer generating function, valid for $\left|r\right|<1$ [31] (§13.5.1)

$${(1-2rs+r^2)}^{-\lambda }=\sum _{n=0}^{\infty }{C}_n^{\left(\lambda \right)}\left(s\right)r^n,$$

we obtain, with

$$r={\displaystyle \frac{t}{\parallel x\parallel }},s=\xi ·\eta ,$$

the expansion

$$u(x-t\eta )=\sum _{n=0}^{\infty }{C}_n^{\left(\lambda \right)}(\xi ·\eta ){\displaystyle \frac{t^n}{{\parallel x\parallel }^{n+d-2}}}.$$

(11)

Comparing the coefficients of $t^n$ in (10) and (11), we obtain

$${\displaystyle \frac{{(-1)}^n}{n!}}{(\eta ·\nabla )}^{n}u\left(x\right)={\displaystyle \frac{1}{{\parallel x\parallel }^{n+d-2}}}{C}_n^{\left(\lambda \right)}(\xi ·\eta ),$$

that is,

$${(-\eta ·\nabla )}^{n}u\left(x\right)=\Gamma (n+1){\displaystyle \frac{1}{{\parallel x\parallel }^{n+d-2}}}{C}_n^{\left(\lambda \right)}(\xi ·\eta ).$$

Finally, replacing $\eta$ by $y/\parallel y\parallel$ and multiplying both sides by ${\parallel y\parallel }^n$ gives

$${(-y·\nabla )}^{n}u\left(x\right)=\Gamma (n+1){\displaystyle \frac{{\parallel y\parallel }^n}{{\parallel x\parallel }^{n+d-2}}}{C}_n^{\left(\lambda \right)}\left({\displaystyle \frac{x}{\parallel x\parallel }}·{\displaystyle \frac{y}{\parallel y\parallel }}\right),$$

which is exactly (9). □

The decomposition of Maxwell’s identity into radial and angular parts makes it especially amenable to analytic continuation. The dependence on the radius appears as a simple homogeneity factor, while the angular dependence is carried by Gegenbauer functions, which possess a natural continuation in the degree parameter. This is precisely the structure exploited below.

3.2. Fractional Maxwell Formula

Lemma 2 shows that integer directional derivatives of the Newtonian kernel are controlled by zonal harmonics. We now prove that the same structure persists for fractional and complex orders. The first proof below is direct and uses only harmonicity, homogeneity, rotational invariance, and the Gegenbauer differential equation. The second proof, given afterward, interprets the same formula through the Ramanujan–Hardy interpolation developed earlier. The two arguments are complementary: the first establishes the analytic identity directly, while the second explains why it fits naturally into the Ramanujan–Hardy framework.

We begin by recalling the standard facts on hypergeometric functions and Gegenbauer functions of complex degree that will be needed for analytic continuation. These are classical and may be found in [22,30,35,36].

For $z\in \mathbb{C}$ and $n\in {\mathbb{N}}_0$, the rising factorial is defined by

$${\left(z\right)}_n:=\left\{\begin{array}{cc}1,\hfill & n=0,\hfill \\ z(z+1)\dots (z+n-1),\hfill & n\ge 1,\hfill \end{array}\right.$$

(12)

and satisfies the identities

$${\left(z\right)}_n={\displaystyle \frac{\Gamma (z+n)}{\Gamma \left(z\right)}},{(z-n)}_n={(-1)}^n{(1-z)}_n.$$

(13)

For parameters $a,b,c\in \mathbb{C}$ with $c\notin \{0,-1,-2,\cdots \}$, the Gauss hypergeometric series

$${}_{2}F_1(a,b;c;x)=\sum _{k=0}^{\infty }{\displaystyle \frac{{\left(a\right)}_k{\left(b\right)}_k}{{\left(c\right)}_{k}k!}}{x}^k$$

(14)

is absolutely convergent for $\left|x\right|<1$, and extends continuously to $x=1$ whenever $\Re (c-a-b)>0$. The classical evaluation

$${}_{2}F_1(a,b;c;1)={\displaystyle \frac{\Gamma \left(c\right)\Gamma (c-a-b)}{\Gamma (c-a)\Gamma (c-b)}},\Re (c-a-b)>0,$$

is standard; see [36] (§15.4).

For fixed $\lambda >0$ and arbitrary complex degree $\nu \in \mathbb{C}$, the Gegenbauer function $C_{\nu }^{\left(\lambda \right)}\left(z\right)$ is defined by the hypergeometric representation [31,37]

$$C_{\nu }^{\left(\lambda \right)}\left(z\right)={\displaystyle \frac{\Gamma (\nu +2\lambda )}{\Gamma \left(2\lambda \right)\Gamma (\nu +1)}}{}_{2}F_1\left(\begin{array}{c}-\nu ,\nu +2\lambda \\ \lambda +\frac{1}{2}\end{array};{\displaystyle \frac{1-z}{2}}\right),z\in [-1,1].$$

(15)

When $\nu \in {\mathbb{N}}_0$, this reduces to the classical Gegenbauer polynomial of degree $\nu$; see [31] (§ 3.15). For each fixed $z\in (-1,1)$, the map $\nu \mapsto C_{\nu }^{\left(\lambda \right)}\left(z\right)$ is analytic in the half-plane

$$\Re \left(\nu \right)>-\lambda ,$$

and admits a meromorphic continuation to the whole complex plane, with simple poles located at

$$\nu =-2\lambda ,-2\lambda -1,-2\lambda -2,\dots$$

(see [37]). In particular, these singularities arise from the Gamma factor $\Gamma {(\nu +1)}^{-1}$ in (15). If $2\lambda \in \mathbb{N}$, then the pole sequence terminates and $C_{\nu }^{\left(\lambda \right)}\left(z\right)$ becomes an entire function of $\nu$. In this case, it satisfies a symmetry (or antisymmetry) relation with respect to $\nu =-\lambda$. Moreover, $C_{\nu }^{\left(\lambda \right)}$ is a solution of the Gegenbauer differential equation

$$(1-z^2)w^{\prime \prime }\left(z\right)-(2\lambda +1)z{w}^{\prime }\left(z\right)+\nu (\nu +2\lambda )w\left(z\right)=0,$$

which characterizes it uniquely (up to normalization) among functions regular at $z=1$. An integral representation, valid for $z=cos\phi$ with $0<\phi <\pi$, is given by [31] (§ 3.15.2)

$$C_{\nu }^{\left(\lambda \right)}(cos\phi )={\displaystyle \frac{2^{\lambda }}{\sqrt{\pi }}}{\displaystyle \frac{\Gamma (\nu +2\lambda )\Gamma (\lambda +\frac{1}{2})}{\Gamma \left(\lambda \right)\Gamma \left(2\lambda \right)\Gamma (\nu +1)}}{(sin\phi )}^{1-2\lambda }{\int }_0^{\phi }cos\left[(\nu +\lambda )\theta \right]{(cos\theta -cos\phi )}^{\lambda -1}d\theta .$$

(16)

Lemma 3.

Let $\lambda >0$. There exists a constant $K_{\lambda }>0$ such that

$$|C_{\nu }^{\lambda }(cos\phi )|\le K_{\lambda }{(sin\phi )}^{-\lambda }{(1+|\nu \left|\right)}^{2\lambda -1}{e}^{\phi |\Im \nu |},0<\phi <\pi .$$

(17)

Proof. 

Fix $\lambda >0$, $\nu \in \mathbb{C}$, and $0<\phi <\pi$. Taking absolute values in (16) gives

$$|C_{\nu }^{\left(\lambda \right)}(cos\phi )|\le A_{\lambda }\left|{\displaystyle \frac{\Gamma (\nu +2\lambda )}{\Gamma (\nu +1)}}\right|{(sin\phi )}^{1-2\lambda }{\int }_0^{\phi }\left|cos\left((\nu +\lambda )\theta \right)\right|{(cos\theta -cos\phi )}^{\lambda -1}d\theta ,$$

(18)

where

$$A_{\lambda }:={\displaystyle \frac{2^{\lambda }}{\sqrt{\pi }}}{\displaystyle \frac{\Gamma \left(\lambda +\frac{1}{2}\right)}{\Gamma \left(\lambda \right)\Gamma \left(2\lambda \right)}}.$$

Since $\lambda >0$ is real, we have

$$\left|cos\left((\nu +\lambda )\theta \right)\right|\le e^{\theta |\Im \nu |}\le e^{\phi |\Im \nu |},0\le \theta \le \phi .$$

Hence

$$|C_{\nu }^{\left(\lambda \right)}(cos\phi )|\le A_{\lambda }\left|{\displaystyle \frac{\Gamma (\nu +2\lambda )}{\Gamma (\nu +1)}}\right|e^{\phi |\Im \nu |}{(sin\phi )}^{1-2\lambda }{\int }_0^{\phi }{(cos\theta -cos\phi )}^{\lambda -1}d\theta .$$

(19)

We now estimate the integral. Using

$$cos\theta -cos\phi =2sin\left({\displaystyle \frac{\theta +\phi }{2}}\right)sin\left({\displaystyle \frac{\phi -\theta }{2}}\right),0\le \theta \le \phi ,$$

and observing that

$$sin\left({\displaystyle \frac{\theta +\phi }{2}}\right)\ge {\displaystyle \frac{1}{2}}sin\phi ,0\le \theta \le \phi ,$$

we obtain

$$cos\theta -cos\phi \ge sin\phi sin\left({\displaystyle \frac{\phi -\theta }{2}}\right).$$

Therefore,

$${(cos\theta -cos\phi )}^{\lambda -1}\le {(sin\phi )}^{\lambda -1}{\left(sin{\displaystyle \frac{\phi -\theta }{2}}\right)}^{\lambda -1}.$$

It follows that

$${\int }_0^{\phi }{(cos\theta -cos\phi )}^{\lambda -1}d\theta \le {(sin\phi )}^{\lambda -1}{\int }_0^{\phi }{\left(sin{\displaystyle \frac{\phi -\theta }{2}}\right)}^{\lambda -1}d\theta .$$

With the change of variables $u=(\phi -\theta )/2$, we get

$${\int }_0^{\phi }{\left(sin{\displaystyle \frac{\phi -\theta }{2}}\right)}^{\lambda -1}d\theta =2{\int }_0^{\phi /2}{(sinu)}^{\lambda -1}du\le 2{\int }_0^{\pi /2}{(sinu)}^{\lambda -1}du=:B_{\lambda }<\infty ,$$

since $\lambda >0$. Thus,

$${\int }_0^{\phi }{(cos\theta -cos\phi )}^{\lambda -1}d\theta \le B_{\lambda }{(sin\phi )}^{\lambda -1}.$$

(20)

Substituting (20) into (19), we obtain

$$|C_{\nu }^{\left(\lambda \right)}(cos\phi )|\le A_{\lambda }{B}_{\lambda }\left|{\displaystyle \frac{\Gamma (\nu +2\lambda )}{\Gamma (\nu +1)}}\right|{(sin\phi )}^{-\lambda }{e}^{\phi |\Im \nu |}.$$

Finally, by the standard Gamma-ratio estimate, there exists a constant $M_{\lambda }>0$ such that

$$\left|{\displaystyle \frac{\Gamma (\nu +2\lambda )}{\Gamma (\nu +1)}}\right|\le M_{\lambda }{(1+|\nu \left|\right)}^{2\lambda -1},\nu \in \mathbb{C}.$$

Therefore,

$$|C_{\nu }^{\left(\lambda \right)}(cos\phi )|\le K_{\lambda }{(sin\phi )}^{-\lambda }{(1+|\nu \left|\right)}^{2\lambda -1}{e}^{\phi |\Im \nu |},$$

where

$$K_{\lambda }:=A_{\lambda }{B}_{\lambda }{M}_{\lambda }.$$

This proves (17). □

We can now prove the fractional analog of Maxwell’s identity.

Proposition 4.

Let $d\ge 3$ and set $\lambda ={\displaystyle \frac{d}{2}}-1$. Let $0<\Re \left(s\right)<1$ and $u\in (-1,1)$. Then,

$${\displaystyle \frac{s}{\Gamma (1-s)}}{\int }_0^{\infty }{\displaystyle \frac{1-{(1+2u\tau +{\tau }^2)}^{-\lambda }}{{\tau }^{1+s}}}d\tau =\Gamma (1+s)C_s^{\left(\lambda \right)}\left(u\right).$$

(21)

Proof. 

Fix $u\in (-1,1)$ and define

$${\varphi }_0\left(z\right):=C_z^{\left(\lambda \right)}\left(u\right),z\in \mathbb{C}.$$

Since $2\lambda =d-2\in {\mathbb{N}}_0$, it follows from that ${\varphi }_0$ is an entire function of z. Moreover, by Lemma 3, there exist constants $C,B,A>0$ with $A<\pi$ such that

$$|{\varphi }_0{(v+iw)|\le C(1+|v+iw|)}^B{e}^{A\left|w\right|},v+iw\in \mathbb{C},$$

so that ${\varphi }_0$ satisfies the Hardy growth condition on every half-plane

$$H\left(\delta \right)=\{z\in \mathbb{C}:\Re z\ge -\delta \},0<\delta <1.$$

For $\left|t\right|<1$, the Gegenbauer generating function gives

$$\sum _{n=0}^{\infty }{C}_n^{\left(\lambda \right)}\left(u\right)t^n={(1-2ut+t^2)}^{-\lambda }.$$

Replacing t by $-t$, we obtain

$$\sum _{n=0}^{\infty }{(-1)}^n{\varphi }_0\left(n\right)t^n=\sum _{n=0}^{\infty }{C}_n^{\left(\lambda \right)}\left(u\right){(-t)}^n={(1+2ut+t^2)}^{-\lambda },\left|t\right|<1.$$

Thus, the function

$$F\left(t\right):={(1+2ut+t^2)}^{-\lambda }$$

admits a local expansion of the form required by the shifted Ramanujan–Hardy theorem (Theorem 3) with interpolant ${\varphi }_0$. Therefore, for $0<\Re \left(s\right)<1$, the theorem yields

$${\int }_0^{\infty }{t}^{-1-s}\left(F\left(t\right)-1\right)dt=-{\displaystyle \frac{\pi }{sin\left(\pi s\right)}}{\varphi }_0\left(s\right)=-{\displaystyle \frac{\pi }{sin\left(\pi s\right)}}{C}_s^{\left(\lambda \right)}\left(u\right).$$

Multiplying both sides by $-s/\Gamma (1-s)$ and using Euler’s reflection identity

$$\Gamma (1+s)\Gamma (1-s)={\displaystyle \frac{\pi s}{sin\left(\pi s\right)}},$$

we obtain

$${\displaystyle \frac{s}{\Gamma (1-s)}}{\int }_0^{\infty }{\displaystyle \frac{1-{(1+2ut+t^2)}^{-\lambda }}{t^{1+s}}}dt=\Gamma (1+s)C_s^{\left(\lambda \right)}\left(u\right),$$

which is exactly (21). □

Theorem 4.

Let $d\ge 3$ and $0<\Re \left(s\right)<1$, and set

$$\lambda ={\displaystyle \frac{d-2}{2}}.$$

Let $x\in {\mathbb{R}}^d\setminus \left\{0\right\}$ and $\eta \in {\mathbb{S}}^{d-1}$ be such that

$${\displaystyle \frac{x}{\parallel x\parallel }}·\eta \in (-1,1).$$

Then

$${(-\eta ·\nabla )}^s\left({\parallel x\parallel }^{2-d}\right)=\Gamma (1+s){\displaystyle \frac{C_s^{\left(\lambda \right)}\left(\frac{x}{\parallel x\parallel }·\eta \right)}{{\parallel x\parallel }^{d+s-2}}}.$$

(22)

Proof. 

Let

$$K\left(x\right):={\parallel x\parallel }^{2-d}={\parallel x\parallel }^{-2\lambda }.$$

Since $u>-1$, the half-line $\{x+t\eta :t\ge 0\}$ does not intersect the origin. Indeed, if $x+t\eta =0$ for some $t\ge 0$, then $x=-t\eta$, which implies $\xi =-\eta$ and hence $u=-1$, a contradiction. Therefore, the map $t\mapsto K(x+t\eta )$ is smooth on $[0,\infty )$. We claim that the Marchaud integral

$${(-\eta ·\nabla )}^{s}K\left(x\right)={\displaystyle \frac{s}{\Gamma (1-s)}}{\int }_0^{\infty }{\displaystyle \frac{K\left(x\right)-K(x+t\eta )}{t^{1+s}}}dt$$

is absolutely convergent.

As $t\to 0^{+}$, since K is smooth on ${\mathbb{R}}^d\setminus \left\{0\right\}$, Taylor’s formula gives, see Lemma 1

$$K(x+t\eta )=K\left(x\right)+t(\eta ·\nabla K)\left(x\right)+O\left(t^2\right),$$

so that

$$|K\left(x\right)-K(x+t\eta )|\le C_{x}t,0<t<1.$$

Hence

$${\displaystyle \frac{\left|K\right(x)-K(x+t\eta \left)\right|}{t^{1+\Re \left(s\right)}}}\le C_x{t}^{-\Re \left(s\right)},$$

which is integrable near 0 since $0<\Re \left(s\right)<1$. For $t\ge 1$, we use

$${\parallel x+t\eta \parallel }^2=r^2+2rtu+t^2={(t+ru)}^2+r^2(1-u^2)\ge r^2(1-u^2)>0,$$

so there exists $c_{x,\eta }>0$ such that

$$\parallel x+t\eta \parallel \ge c_{x,\eta }(1+t),t\ge 1.$$

It follows that

$$\left|K(x+t\eta )\right|\le C_{x,\eta }{(1+t)}^{-2\lambda }.$$

Therefore,

$${\displaystyle \frac{\left|K\right(x)-K(x+t\eta \left)\right|}{t^{1+\Re \left(s\right)}}}\le {\displaystyle \frac{\left|K\right(x\left)\right|}{t^{1+\Re \left(s\right)}}}+{\displaystyle \frac{C_{x,\eta }}{t^{1+\Re \left(s\right)}{(1+t)}^{2\lambda }}},$$

which is integrable on $(1,\infty )$ since $\Re \left(s\right)>0$ and $\lambda >0$. Thus, the Marchaud integral converges absolutely.

Using

$$K(x+t\eta )={(r^2+2rtu+t^2)}^{-\lambda },$$

we obtain

$${(-\eta ·\nabla )}^{s}K\left(x\right)={\displaystyle \frac{s}{\Gamma (1-s)}}{\int }_0^{\infty }{\displaystyle \frac{r^{-2\lambda }-{(r^2+2rtu+t^2)}^{-\lambda }}{t^{1+s}}}dt.$$

Since the integral is absolutely convergent, the change of variables $t=r\tau$ is justified. This yields

$$r^2+2rtu+t^2=r^2(1+2u\tau +{\tau }^2),$$

and therefore

$${(r^2+2rtu+t^2)}^{-\lambda }=r^{-2\lambda }{(1+2u\tau +{\tau }^2)}^{-\lambda }.$$

Substituting gives

$${(-\eta ·\nabla )}^{s}K\left(x\right)=r^{-2\lambda -s}{\displaystyle \frac{s}{\Gamma (1-s)}}{\int }_0^{\infty }{\displaystyle \frac{1-{(1+2u\tau +{\tau }^2)}^{-\lambda }}{{\tau }^{1+s}}}d\tau .$$

By Proposition 4,

$${\displaystyle \frac{s}{\Gamma (1-s)}}{\int }_0^{\infty }{\displaystyle \frac{1-{(1+2u\tau +{\tau }^2)}^{-\lambda }}{{\tau }^{1+s}}}d\tau =\Gamma (1+s)C_s^{\left(\lambda \right)}\left(u\right).$$

Hence

$${(-\eta ·\nabla )}^{s}K\left(x\right)=\Gamma (1+s)r^{-2\lambda -s}{C}_s^{\left(\lambda \right)}\left(u\right).$$

Since $u=\xi ·\eta$ and $2\lambda +s=d+s-2$, we obtain

$${(-\eta ·\nabla )}^s\left({\parallel x\parallel }^{2-d}\right)=\Gamma (1+s){\displaystyle \frac{C_s^{\left(\lambda \right)}(\xi ·\eta )}{{\parallel x\parallel }^{d+s-2}}},$$

which proves (22). □

Proposition 5.

Let

$${\mathrm{\Omega }}_{\eta }:={\mathbb{R}}^d\setminus \{t\eta :t\ge 0\}.$$

Let $d\ge 3$, let $0<\Re s<1$, and let

$$K\left(x\right):={\parallel x\parallel }^{2-d},x\in {\mathbb{R}}^d\setminus \left\{0\right\}.$$

For $x\in {\mathrm{\Omega }}_{\eta }$, define

$$F_s\left(x\right):={(-\eta ·\nabla )}^{s}K\left(x\right):={\displaystyle \frac{s}{\Gamma (1-s)}}{\int }_0^{\infty }{\displaystyle \frac{K\left(x\right)-K(x+t\eta )}{t^{1+s}}}dt.$$

Then $F_s$ is well-defined on ${\mathrm{\Omega }}_{\eta }$, belongs to $C^{\infty }\left({\mathrm{\Omega }}_{\eta }\right)$, and satisfies

$$\Delta F_s=0\mathrm{in}{\mathrm{\Omega }}_{\eta }.$$

Proof. 

Fix a compact set $E⋐{\mathrm{\Omega }}_{\eta }$. Since E is disjoint from the closed ray $\{t\eta :t\ge 0\}$, there exists $\delta >0$ such that

$$\parallel x+t\eta \parallel \ge \delta ,x\in E,t\ge 0.$$

(23)

We first we prove smoothness. Let $\alpha$ be any multi-index. Since ${\partial }^{\alpha }K$ is smooth on ${\mathbb{R}}^d\setminus \left\{0\right\}$, then for $0<t\le 1$,

$$|{\partial }^{\alpha }K\left(x\right)-{\partial }^{\alpha }K(x+t\eta )|\le C_{\alpha ,E}t,x\in E,$$

because $\nabla {\partial }^{\alpha }K$ is bounded on the corresponding compact set. For $t\ge 1$,

$$|{\partial }^{\alpha }K(x+t\eta )|\le C_{\alpha ,E}{(1+t)}^{2-d-\left|\alpha \right|},x\in E,$$

and ${\partial }^{\alpha }K\left(x\right)$ is bounded on E. Therefore,

$$\left|{\displaystyle \frac{{\partial }^{\alpha }K\left(x\right)-{\partial }^{\alpha }K(x+t\eta )}{t^{1+s}}}\right|\le g_{\alpha ,E}\left(t\right),x\in E,t>0,$$

for some $g_{\alpha ,E}\in L^1(0,\infty )$. Hence, differentiation under the integral sign is justified by dominated convergence, and

$${\partial }^{\alpha }{F}_s\left(x\right)={\displaystyle \frac{s}{\Gamma (1-s)}}{\int }_0^{\infty }{\displaystyle \frac{{\partial }^{\alpha }K\left(x\right)-{\partial }^{\alpha }K(x+t\eta )}{t^{1+s}}}dt,x\in E.$$

Since $\alpha$ is arbitrary, $F_s\in C^{\infty }\left(E\right)$, and therefore $F_s\in C^{\infty }\left({\mathrm{\Omega }}_{\eta }\right)$. Finally, taking $\left|\alpha \right|=2$ and summing over the coordinate directions yields

$$\Delta F_s\left(x\right)={\displaystyle \frac{s}{\Gamma (1-s)}}{\int }_0^{\infty }{\displaystyle \frac{\Delta K\left(x\right)-\Delta K(x+t\eta )}{t^{1+s}}}dt.$$

For $x\in {\mathrm{\Omega }}_{\eta }$ and $t\ge 0$, both x and $x+t\eta$ are nonzero; indeed, $x+t\eta =0$ would imply $x=-t\eta \in \{r\eta :r\le 0\}$, which is excluded by the definition of ${\mathrm{\Omega }}_{\eta }$. Since K is harmonic on ${\mathbb{R}}^d\setminus \left\{0\right\}$, we have

$$\Delta K\left(x\right)=0,\Delta K(x+t\eta )=0.$$

Hence, the integrand vanishes identically, so

$$\Delta F_s\left(x\right)=0,x\in {\mathrm{\Omega }}_{\eta }.$$

This proves the claim. □

Proof of Theorem 4.

Let

$$K\left(x\right):={\parallel x\parallel }^{2-d}={\parallel x\parallel }^{-2\lambda },\lambda ={\displaystyle \frac{d-2}{2}},$$

and define

$$F_s\left(x\right):={(-\eta ·\nabla )}^{s}K\left(x\right),x\in {\mathbb{R}}^d\setminus \left\{0\right\}.$$

By Proposition 1, the function $F_s$ is homogeneous of degree $-(2\lambda +s)$ and invariant under rotations fixing $\eta$. Hence, $F_s\left(x\right)$ depends only on

$$r=\parallel x\parallel ,u:={\displaystyle \frac{x}{\parallel x\parallel }}·\eta ,$$

and there exists a function $G_s:(-1,1)\to \mathbb{C}$ such that

$$F_s\left(x\right)=r^{-2\lambda -s}{G}_s\left(u\right).$$

For $\left|u\right|<1$, one has $x\in {\mathrm{\Omega }}_{\eta }:={\mathbb{R}}^d\setminus \{-t\eta :t\ge 0\}$, and by Proposition 5, the function $F_s$ is harmonic at x. Substituting the ansatz into the identity for the Laplacian of functions of the form $r^{\alpha }g\left(u\right)$,

$$\Delta \left(r^{\alpha }g\left(u\right)\right)=r^{\alpha -2}\left[(1-u^2)g^{\prime \prime }\left(u\right)-(2\lambda +1)u{g}^{\prime }\left(u\right)+\alpha (\alpha +2\lambda )g\left(u\right)\right],$$

with $\alpha =-(2\lambda +s)$, we obtain

$$(1-u^2)G_s^{\prime \prime }\left(u\right)-(2\lambda +1)u{G}_s^{\prime }\left(u\right)+s(s+2\lambda )G_s\left(u\right)=0,-1<u<1.$$

Thus, $G_s$ satisfies the Gegenbauer differential equation of degree s and parameter $\lambda$. Since $F_s$ is smooth on ${\mathbb{R}}^d\setminus \left\{0\right\}$, the function $G_s$ is regular on $(-1,1)$ and bounded near $u=1$, hence

$$G_s\left(u\right)=c\left(s\right)C_s^{\left(\lambda \right)}\left(u\right)$$

for some constant $c\left(s\right)$. Therefore,

$$F_s\left(x\right)=c\left(s\right){\displaystyle \frac{C_s^{\left(\lambda \right)}\left(\frac{x}{\parallel x\parallel }·\eta \right)}{{\parallel x\parallel }^{d+s-2}}},\left|u\right|<1.$$

To determine the constant $c\left(s\right)$, we evaluate $F_s$ along the axis $x=r\eta$, $r>0$, using the definition of the fractional directional derivative.

By the Marchaud formula,

$$F_s\left(r\eta \right)={(-\eta ·\nabla )}^{s}K\left(r\eta \right)={\displaystyle \frac{s}{\Gamma (1-s)}}{\int }_0^{\infty }{\displaystyle \frac{K\left(r\eta \right)-K(r\eta +t\eta )}{t^{1+s}}}dt.$$

Since $K\left(r\eta \right)=r^{-2\lambda }$ and $K(r\eta +t\eta )={(r+t)}^{-2\lambda }$, this gives

$$F_s\left(r\eta \right)={\displaystyle \frac{s}{\Gamma (1-s)}}{\int }_0^{\infty }{\displaystyle \frac{r^{-2\lambda }-{(r+t)}^{-2\lambda }}{t^{1+s}}}dt.$$

Using the Laplace representation

$$r^{-2\lambda }={\displaystyle \frac{1}{\Gamma \left(2\lambda \right)}}{\int }_0^{\infty }{a}^{2\lambda -1}{e}^{-ar}da,$$

Substituting this into the Marchaud integral and using Fubini’s theorem (justified by absolute convergence), we obtain

$$F_s\left(r\eta \right)={\displaystyle \frac{1}{\Gamma \left(2\lambda \right)}}{\int }_0^{\infty }{a}^{2\lambda -1}\left[{\displaystyle \frac{s}{\Gamma (1-s)}}{\int }_0^{\infty }{\displaystyle \frac{e^{-ar}-e^{-a(r+t)}}{t^{1+s}}}dt\right]da.$$

The inner integral is the one-dimensional Marchaud fractional derivative of the function $r\mapsto e^{-ar}$, and it is well-known that

$${\displaystyle \frac{s}{\Gamma (1-s)}}{\int }_0^{\infty }{\displaystyle \frac{e^{-ar}-e^{-a(r+t)}}{t^{1+s}}}dt=a^s{e}^{-ar}.$$

Therefore,

$$F_s\left(r\eta \right)={\displaystyle \frac{1}{\Gamma \left(2\lambda \right)}}{\int }_0^{\infty }{a}^{2\lambda +s-1}{e}^{-ar}da={\displaystyle \frac{\Gamma (2\lambda +s)}{\Gamma \left(2\lambda \right)}}{r}^{-2\lambda -s}.$$

On the other hand, from the structural representation,

$$F_s\left(r\eta \right)=c\left(s\right)C_s^{\left(\lambda \right)}\left(1\right)r^{-2\lambda -s}.$$

Using the identity

$$C_s^{\left(\lambda \right)}\left(1\right)={\displaystyle \frac{\Gamma (s+2\lambda )}{\Gamma \left(2\lambda \right)\Gamma (s+1)}},$$

we obtain

$$c\left(s\right){\displaystyle \frac{\Gamma (s+2\lambda )}{\Gamma \left(2\lambda \right)\Gamma (s+1)}}={\displaystyle \frac{\Gamma (s+2\lambda )}{\Gamma \left(2\lambda \right)}},$$

and hence

$$c\left(s\right)=\Gamma (1+s).$$

Consequently,

$${(-\eta ·\nabla )}^s\left({\parallel x\parallel }^{2-d}\right)=\Gamma (1+s){\displaystyle \frac{C_s^{\left(\lambda \right)}\left(\frac{x}{\parallel x\parallel }·\eta \right)}{{\parallel x\parallel }^{d+s-2}}},\left|u\right|<1.$$

By continuity, the identity extends to $u=\pm 1$, which completes the proof. □

Theorem 4 provides an analytic extension of the classical Maxwell identity from integer orders to complex orders s with $0<\Re \left(s\right)<1$. Namely, the formula

$${(-\eta ·\nabla )}^n\left({\parallel x\parallel }^{2-d}\right)=\Gamma (n+1){\displaystyle \frac{C_n^{\left(\lambda \right)}(\xi ·\eta )}{{\parallel x\parallel }^{d+n-2}}},n\in {\mathbb{N}}_0,$$

extends to non-integer s by replacing n with s and $C_n^{\left(\lambda \right)}$ with $C_s^{\left(\lambda \right)}$.

This extension preserves the structural features of the classical identity: it remains homogeneous of degree $-(d+s-2)$ and depends on the angular variable only through $\xi ·\eta$. Moreover, away from the singular ray $\{-t\eta :t\ge 0\}$, the resulting function is harmonic, reflecting the underlying elliptic structure despite the nonlocal character of the fractional operator. This result does not follow from a global Taylor expansion along the ray $x+t\eta$. Indeed, the series

$$K(x+t\eta )=\sum _{n=0}^{\infty }{\displaystyle \frac{t^n}{n!}}{(\eta ·\nabla )}^{n}K\left(x\right),K\left(x\right)={\parallel x\parallel }^{2-d},$$

is only locally valid for $\left|t\right|<\parallel x\parallel$ and cannot be used to justify the fractional identity globally. This distinction between local expansions and global identities is essential for the validity of the argument. The appropriate framework is instead provided by analytic interpolation. For $2\lambda \in \mathbb{N}$, the function

$$\varphi (z,x):={\displaystyle \frac{C_z^{\left(\lambda \right)}(\xi ·\eta )}{{\parallel x\parallel }^{d+z-2}}}$$

is entire in z and satisfies a Hardy-type growth condition of exponential type strictly less than $\pi$. By the classical Maxwell identity,

$$\varphi (n,x)={\displaystyle \frac{1}{n!}}{(-\eta ·\nabla )}^{n}K\left(x\right),n\in {\mathbb{N}}_0,$$

so $\varphi$ interpolates the directional derivatives at integer orders. Under the stated growth condition, such an interpolant is unique in the sense of Carlson–Boas type theorems, which ensures that the corresponding Ramanujan–Hardy fractional derivative is well defined. Ramanujan’s Master Theorem then yields an analytic function of s given by

$${(-\eta ·\nabla )}_{\mathrm{RH}}^{s}K\left(x\right)=\Gamma (1+s)\varphi (s,x),0<\Re \left(s\right)<1.$$

Thus, the fractional Maxwell identity is best understood as an analytic interpolation of the classical formula: the discrete family of directional derivatives extends uniquely to complex order, and the Gegenbauer function $C_s^{\left(\lambda \right)}$ appears as the natural continuation of the zonal harmonic coefficients.