Summed Series Involving ₁F₂ Hypergeometric Functions

Abstract

Summation of infinite series has played a significant role in a broad range of problems in the physical sciences and is of interest in a purely mathematical context. In a prior paper, we found that the Fourier–Legendre series of a Bessel function of the first kind $J_N\left(kx\right)$ and modified Bessel functions of the first kind $I_N\left(kx\right)$ lead to an infinite set of series involving ${}_{1}F_2$ hypergeometric functions (extracted therefrom) that could be summed, having values that are inverse powers of the eight primes $1/\left(2^i{3}^j{5}^k{7}^l{11}^m{13}^n{17}^o{19}^p\right)$ multiplying powers of the coefficient k, for the first 22 terms in each series. The present paper shows how to generate additional, doubly infinite summed series involving ${}_{1}F_2$ hypergeometric functions from Chebyshev polynomial expansions of Bessel functions, and trebly infinite sets of summed series involving ${}_{1}F_2$ hypergeometric functions from Gegenbauer polynomial expansions of Bessel functions. That the parameters in these new cases can be varied at will significantly expands the landscape of applications for which they could provide a solution.

1. Introduction

The summing of infinite series has played a key part in a broad range of problems in the physical sciences, from self-energy diagrams [1,2,3] to polarization [4]. See [5] for an excellent review of the summation of divergent asymptotic expansions. In particular, Mera et al. [6,7] and Pedersen et al. [8] use hypergeometric functions to sum series in perturbation theory.

In the present paper, we focus on summing trebly infinite sets of series involving ${}_{1}F_2$ hypergeometric functions. Historical antecedents of similar work include Chaundy [9], who expressed products of ${}_{0}F_1$ functions as infinite sums of ${}_{2}F_1$ functions, products of ${}_{1}F_1$ functions and products of ${}_{2}F_0$ functions as infinite sums of ${}_{3}F_2$ functions, and products of ${}_{2}F_1$ functions as infinite sums of ${}_{4}F_3$ functions. Additional combinations are found in Burchnall and Chaundy [10], Henrici [11], Gasper [12], and Jain and Verma [13].

Slater expressed generalized Whittaker functions [14] (${}_{p}F_p$ functions having $p\ge 1$) as sums of other generalized Whittaker functions, and a few other ${}_{p}F_q$ functions. Additional forms of the latter are found in [15,16,17,18,19,20,21].

Of course, the Appell functions of the first through fourth kinds are defined [22] as infinite sums over ${}_{2}F_1$ functions, while some Meijer G-functions [23] and certain Kampé de Fériet’s functions may be expressed [24] as infinite sums over ${}_{p}F_q$ functions.

The final stop in this historical sketch is the expression of ${}_{p}F_q$ functions as finite sums of other ${}_{p}F_q$ functions [25] that act as recurrence relations [26].

In a prior paper [27], I refined Keating’s [28] derivation of the coefficient set of the Fourier–Legendre series for the Bessel function $J_N\left(kx\right)$ to be

$$J_N\left(kx\right)=\sum _{L=0}^{\infty }{a}_{LN}\left(k\right)P_L\left(x\right)$$

(1)

where

$$\begin{array}{ccc}{a}_{LN}\left(k\right)& =& \sqrt{\pi }(2L+1)2^{-L-1}{i}^{L-\mathrm{N}}\sum _{M=0}^{\infty }{\displaystyle \frac{\left({\left(-\frac{1}{4}\right)}^M{k}^{L+2M}\right)}{2^{L+2M+1}\left(M!\mathsf{\Gamma }\left(L+M+\frac{3}{2}\right)\right)}}\\ & \times & \left(1+{(-1)}^{L+2M+\mathrm{N}}\right)\left({\displaystyle \genfrac{}{}{0pt}{}{L+2M}{\frac{1}{2}(L+2M-\mathrm{N})}}\right)\\ & =& {\displaystyle \frac{\sqrt{\pi }{2}^{-2L-2}(2L+1)k^L{i}^{L-N}}{\mathsf{\Gamma }\left(\frac{1}{2}(2L+3)\right)}}\left(1+{(-1)}^{L+N}\right)\left({\displaystyle \genfrac{}{}{0pt}{}{L}{\frac{L-N}{2}}}\right)\\ & \times & {}_{2}F_3\left({\displaystyle \frac{L}{2}}+{\displaystyle \frac{1}{2}},{\displaystyle \frac{L}{2}}+1;L+{\displaystyle \frac{3}{2}},{\displaystyle \frac{L}{2}}-{\displaystyle \frac{N}{2}}+1,{\displaystyle \frac{L}{2}}+{\displaystyle \frac{N}{2}}+1;-{\displaystyle \frac{k^2}{4}}\right)\\ & =& \sqrt{\pi }{2}^{-2L-2}(2L+1)k^L{i}^{L-N}\left(1+{(-1)}^{L+N}\right)\mathsf{\Gamma }(L+1)\\ & \times & {}_2\tilde{F}_3\left({\displaystyle \frac{L}{2}}+{\displaystyle \frac{1}{2}},{\displaystyle \frac{L}{2}}+1;L+{\displaystyle \frac{3}{2}},{\displaystyle \frac{L}{2}}-{\displaystyle \frac{N}{2}}+1,{\displaystyle \frac{L}{2}}+{\displaystyle \frac{N}{2}}+1;-{\displaystyle \frac{k^2}{4}}\right)\end{array},$$

(2)

of which the final two steps were new in the prior work [27]. I included the final form using regularized hypergeometric functions [29]

$${}_{2}F_3\left(a_1,a_2;b_1,b_2,b_3;z\right)=\mathsf{\Gamma }\left(b_1\right)\mathsf{\Gamma }\left(b_2\right)\mathsf{\Gamma }\left(b_3\right){}_2\tilde{F}_3\left(a_1,a_2;b_1,b_2,b_3;z\right)$$

(3)

and canceled the $\mathsf{\Gamma }\left(b_i\right)$ with gamma functions in the denominators of the prefactors. This cancellation allows one to avoid infinities that arise whenever $N>1$ is an integer larger than L, and of the same parity, which would otherwise result in indeterminacies in a computation when one tries to use the conventional form of the hypergeometric function.

After a further review of the literature, I found that Keating’s result (the first line above) and my prior work (the second line above) are implicitly subsumed within Jet Wimp’s 1962 Jacobi expansion [30] of the Anger–Weber function (his equations (2.10) and (2.11)) since Legendre polynomials are a subset of Jacobi polynomials, and the Bessel function $J_N\left(kx\right)$ is a special case of the Anger–Weber function ${\mathbf{J}}_{\nu }\left(kx\right)$ when $\nu$ is an integer. Wimp does not mention the calculational difficulties that were resolved through the third form above.

For the special cases of $N=0,1$, the order of the hypergeometric functions is reduced since the parameters are $a_2=b_3$ and $a_1=b_2$, respectively, giving

$$\begin{array}{ccc}{a}_{L0}\left(k\right)& =& {\displaystyle \frac{\sqrt{\pi }{i}^L{2}^{-2L-2}(2L+1)k^L}{\mathsf{\Gamma }\left(\frac{1}{2}(2L+3)\right)}}\left(1+{(-1)}^L\right)\left({\displaystyle \genfrac{}{}{0pt}{}{L}{\frac{L}{2}}}\right)\\ & \times & {}_{1}F_2\left({\displaystyle \frac{L}{2}}+{\displaystyle \frac{1}{2}};{\displaystyle \frac{L}{2}}+1,L+{\displaystyle \frac{3}{2}};-{\displaystyle \frac{k^2}{4}}\right)\\ & =& \sqrt{\pi }{i}^L{2}^{-2L-2}(2L+1)k^L\mathsf{\Gamma }\left({\displaystyle \frac{L}{2}}+1\right)\left(1+{(-1)}^L\right)\left({\displaystyle \genfrac{}{}{0pt}{}{L}{\frac{L}{2}}}\right)\\ & \times & {}_1\tilde{F}_2\left({\displaystyle \frac{L}{2}}+{\displaystyle \frac{1}{2}};{\displaystyle \frac{L}{2}}+1,L+{\displaystyle \frac{3}{2}};-{\displaystyle \frac{k^2}{4}}\right)\end{array},$$

(4)

and

$$\begin{array}{ccc}{a}_{L1}\left(k\right)& =& {\displaystyle \frac{\sqrt{\pi }{i}^{L-1}{2}^{-2L-2}(2L+1)k^L}{\mathsf{\Gamma }\left(\frac{1}{2}(2L+3)\right)}}\left(1+{(-1)}^{L+1}\right)\left({\displaystyle \genfrac{}{}{0pt}{}{L}{\frac{L-1}{2}}}\right)\\ & \times & {}_{1}F_2\left({\displaystyle \frac{L}{2}}+1;{\displaystyle \frac{L}{2}}+{\displaystyle \frac{3}{2}},L+{\displaystyle \frac{3}{2}};-{\displaystyle \frac{k^2}{4}}\right)\\ & =& i^{L-1}{2}^{-L-2}(2L+1)k^L\mathsf{\Gamma }\left({\displaystyle \frac{L}{2}}+1\right)\left(1+{(-1)}^{L+1}\right)\\ & \times & {}_1\tilde{F}_2\left({\displaystyle \frac{L}{2}}+1;{\displaystyle \frac{L}{2}}+{\displaystyle \frac{3}{2}},L+{\displaystyle \frac{3}{2}};-{\displaystyle \frac{k^2}{4}}\right)\end{array}.$$

(5)

In each special case, the first form involving a hypergeometric function has no numerical indeterminacies, but I include the regularized hypergeometric function version for completeness.

The first 22 terms in the Fourier–Legendre series for $J_0\left(kx\right)$ (1) are given in Appendix A, with $k=1$, as is an updated polynomial approximation created by expanding the Legendre polynomials into their constituent terms and gathering like powers. Since each Legendre polynomial in (A1) contributes to the constant term in both (A2) and (A3), their sum is

$$\begin{array}{cc}\hfill 0.919730410089760239314421194080620\times & \left(1\right)\\ \hfill -0.157942058625851887573713967144364\times & {\displaystyle \frac{1}{2}}{\left(3x^2-1\right)}_{x\to 0}\\ \hfill +0.00343840094460110923299688787207292\times & {\displaystyle \frac{1}{8}}{\left(35x^4-30x^2+3\right)}_{x\to 0}\\ \hfill -0.0000291972184882872969366059098612566\times & {\displaystyle \frac{1}{16}}{\left(231x^6-315x^4+105x^2-5\right)}_{x\to 0}\\ \hfill +\cdots & =& 1\hfill \end{array}$$

(6)

rather than some other number close to 1. This may formalized in a theorem for these summed series:

Theorem 1.

For integer h and for any values of k,

$$\begin{array}{ccc}\hfill \sum _{L=0}^{\infty }& & {\displaystyle \frac{\sqrt{\pi }{i}^L{2}^{-2L-2}\left(1+{(-1)}^L\right)(2L+1)\left(\genfrac{}{}{0pt}{}{L}{\frac{L}{2}}\right)}{\mathsf{\Gamma }\left(\frac{1}{2}(2L+3)\right)}}{}_{1}F_2\left({\displaystyle \frac{L}{2}}+{\displaystyle \frac{1}{2}};{\displaystyle \frac{L}{2}}+1,L+{\displaystyle \frac{3}{2}};-{\displaystyle \frac{k^2}{4}}\right)k^L\hfill \\ & \times & \left\{{\displaystyle \frac{i^L{2}^{-L/2}(L-1)!!{\left(\frac{L}{2}+\frac{1}{2}\right)}_h{\left(-\frac{L}{2}\right)}_h}{h!\frac{L}{2}!{\left(\frac{1}{2}\right)}_h}}\right\}={\displaystyle \frac{{(-1)}^h{2}^{-2h}}{h!\mathsf{\Gamma }(h+1)}}{k}^{2h},\hfill \end{array}$$

(7)

within which $h=0$ gives (6).

A researcher seeking to sum a series like this is likely to have the various factors expressed in alternative ways. For instance, the expression $\left(1+{(-1)}^L\right)$ in the first factor of this equation restricts the sum to even values of L, which is sometimes indicated instead as ${{\sum }_{L=0}^{\infty }}^{\left(2\right)}\dots$. This restriction also means that the double factorial in the next line can be alternatively expressed as $(L-1)!!={\displaystyle \frac{2^{L/2}\mathsf{\Gamma }\left(\frac{L}{2}+\frac{1}{2}\right)}{\sqrt{\pi }}}$. The binomial $\left({\displaystyle \genfrac{}{}{0pt}{}{L}{\frac{L}{2}}}\right)$ can alternatively be expressed as a ratio of gamma functions, ${\displaystyle \frac{\mathsf{\Gamma }(L+1)}{\mathsf{\Gamma }{\left(\frac{L}{2}+1\right)}^2}}$, as can the Pochhammer symbols ${\left(a\right)}_h={\displaystyle \frac{\mathsf{\Gamma }(a+h)}{\mathsf{\Gamma }\left(a\right)}}$. In Equation (26) of the previous paper [27], the term in curly brackets was given as

$$\left\{{\displaystyle \frac{2^{-L}\left(\genfrac{}{}{0pt}{}{2L}{L}\right){\left(\frac{1}{2}-\frac{L}{2}\right)}_{\frac{L}{2}-h}{\left(-\frac{L}{2}\right)}_{\frac{L}{2}-h}}{\left(\frac{L}{2}-h\right)!{\left(\frac{1}{2}-L\right)}_{\frac{L}{2}-h}}}\right\}$$

(8)

because it used an alternative conversion of Legendre polynomials into ${}_{2}F_1$ hypergeometric functions [31] (p. 1044 No. 8911.1) [32] (p. 468 No. 7.3.1.206).

This was proved in the prior work for general h by extracting specific powers of x from the Legendre polynomials, most easily by converting them into ${}_{2}F_1$ hypergeometric functions [31] (p. 1044 No. 8911.2) [32] (p. 466 No. 7.3.1.182) and thence into a finite sum over ratios of Pochhammer symbols. For $h=1$, the $P_2\left(x\right)$ through $P_{42}\left(x\right)$ terms add to give $-1/4$, the coefficient of $x^2$ term in both (A2) and (A3) if $k=1$.

Including $k\ne 1$ poses no problem in (7) despite its appearance as the argument of the ${}_{1}F_2\left({\displaystyle \frac{L}{2}}+{\displaystyle \frac{1}{2}};{\displaystyle \frac{L}{2}}+1,L+{\displaystyle \frac{3}{2}};-{\displaystyle \frac{k^2}{4}}\right)$ function, as well as the existence of a $k^L$ factor in the argument of the sum. It ends up contributing a very clean factor of $k^{2h}$ to the right-hand side of (7). I, thus, summed an infinite set of infinite sums of ${}_{1}F_2$ hypergeometric functions, though I numerically verified only those with $0\le h\le 42$ (I had to take the upper limit on the number of terms in the series ≥$h+74$ in order to obtain a percent difference between left- and right-hand sides that was ≤${10}^{-33}$, because the first h terms in the series do not contribute. For $h=0$, an upper limit on the number of terms in the series ≥$h+44$ was sufficient).

I likewise summed an infinite set of infinite sums of ${}_{1}F_2$ hypergeometric functions derived from the Fourier–Legendre series for $J_1\left(kx\right)$ (1):

Theorem 2.

For integer h and for any values of k,

$$\begin{array}{ccc}\hfill \sum _{L=1}^{\infty }& & {\displaystyle \frac{\sqrt{\pi }{i}^{L-1}\left(1+{(-1)}^{L+1}\right)(2L+1)2^{-3L-2}\left(\genfrac{}{}{0pt}{}{L}{\frac{L-1}{2}}\right)\left(\genfrac{}{}{0pt}{}{2L}{L}\right){(-1)}^{-h+\frac{L}{2}-\frac{1}{2}}}{\mathsf{\Gamma }\left(\frac{1}{2}(2L+3)\right)\left(-h+\frac{L}{2}-\frac{1}{2}\right)!}}{k}^L\hfill \\ & \times & {}_{1}F_2\left({\displaystyle \frac{L}{2}}+1;{\displaystyle \frac{L}{2}}+{\displaystyle \frac{3}{2}},L+{\displaystyle \frac{3}{2}};-{\displaystyle \frac{k^2}{4}}\right)\left[{\displaystyle \frac{{\left(\frac{1}{2}-\frac{L}{2}\right)}_{\frac{L-1}{2}-h}{\left(-\frac{L}{2}\right)}_{\frac{L-1}{2}-h}}{{\left(\frac{1}{2}-L\right)}_{\frac{L-1}{2}-h}}}\right]\hfill \\ & =& {\displaystyle \frac{{(-1)}^h{2}^{-2h-1}}{h!\mathsf{\Gamma }(h+2)}}{k}^{2h+1}.\hfill \end{array}$$

(9)

The question naturally arises as to whether one can derive such a summed infinite series based on other polynomial expansions. In the following, one may answer in the affirmative for both Chebyshev and Gegenbauer polynomial expansions of Bessel functions.

2. Summed Series Involving ${}_{\mathbf{1}}\mathit{F}_{\mathbf{2}}$ Hypergeometric Functions from Chebyshev Polynomial Expansions of Bessel Functions

We wish to prove the following theorem for the summed series derived from Chebyshev polynomial expansions of the $J_0\left(kx\right)$ Bessel function:

Theorem 3.

For integer h and for any values of k,

$$\begin{array}{ccc}\hfill \sum _{L=0}^{\infty }& & {\displaystyle \frac{{(-1)}^L{2}^{-2L}{\left(\frac{1}{2}-L\right)}_{L-h}{(-L)}_{L-h}}{L!\mathsf{\Gamma }(L+1)(L-h)!{(1-2L)}_{L-h}}}{}_{1}F_2\left(L+{\displaystyle \frac{1}{2}};L+1,2L+1;-{\displaystyle \frac{k^2}{4}}\right)k^{2L}\hfill \\ & =& {\displaystyle \frac{{(-1)}^h{2}^{-2h}}{h!\mathsf{\Gamma }(h+1)}}{k}^{2h}.\hfill \end{array}$$

(10)

Proof of Theorem 3.

Wimp also applied his Jacobi expansion [30] to find Chebyshev polynomial expansions of Bessel functions, since [31] (p. 1060 No. 8.962.3)

$$P_{2n}^{\left(-{\displaystyle \frac{1}{2}},-{\displaystyle \frac{1}{2}}\right)}\left(z\right)={\displaystyle \frac{{\left(\frac{1}{2}\right)}_{2n}}{\left(2n\right)!}}{T}_{2n}\left(z\right).$$

(11)

Unlike the section above, the following expansion (his Equations. (3.6) and (3.7)) applies to non-integer indices as well:

$$J_{\nu }\left(kx\right)={\left(kx\right)}^{\nu }\sum _{L=0}^{\infty }{C}_{L\nu }\left(k\right)T_{2L}\left(x\right).-1\le x\le 1$$

(12)

Since what one is expanding in Chebyshev polynomials is the function $J_{\nu }\left(kx\right){\left(kx\right)}^{-\nu }$, the coefficients can only be given by the orthogonality of the Chebyshev polynomials if we include the full function in the defining integral,

$$C_{L\nu }\left(k\right)={\displaystyle \frac{\left(2-{\delta }_{L0}\right)}{2}}{\displaystyle \frac{2}{\pi }}{\int }_{-1}^1\left(J_{\nu }\left(kx\right){\left(kx\right)}^{-\nu }\right){\displaystyle \frac{1}{\sqrt{1-x^2}}}{T}_{2L}\left(x\right)dx,$$

(13)

which Wimp finds to be

$$\begin{array}{ccc}{C}_{L\nu }\left(k\right)& =& {\displaystyle \frac{{(-1)}^L{k}^{2L}{2}^{-4L-\nu }\left(2-{\delta }_{L0}\right)}{L!\mathsf{\Gamma }(L+\nu +1)}}{}_{1}F_2\left(L+{\displaystyle \frac{1}{2}};L+\nu +1,2L+1;-{\displaystyle \frac{k^2}{4}}\right)\end{array}.$$

(14)

The first 22 terms in the Chebyshev polynomial expansion of $J_0\left(kx\right)$ (12), with $k=1$ and 8, are given in Appendix A.2.

Since the constant term of every Chebyshev polynomial has magnitude one, and alternating sign, the sum of these times the coefficients—$a_{2r}$ in Clenshaw’s convention in which sums having a single prime indicate that the term with suffix zero is to be halved—is simply

$${\sum }^{\prime }{\left(-1\right)}^L{a}_{2r}=1.$$

(15)

Clenshaw has, thus, given the first of the summation rules we wish to derive.

The general-h proof follows that of the prior paper. In order to sum the infinite set of infinite sums of ${}_{1}F_2$ hypergeometric functions derived from the Fourier–Legendre series expansion of Bessel functions (7) and (9), we extracted specific powers of x by converting Legendre polynomials into ${}_{2}F_1$ hypergeometric functions [32] (p. 468 No. 7.3.1.206). The equivalent conversion for Chebyshev polynomials is

$$T_{2L}\left(x\right)=2^{2L-1}{x}^{2L}{}_{2}F_1\left({\displaystyle \frac{1}{2}}-L,-L;1-2L;{\displaystyle \frac{1}{x^2}}\right)\left[1+{\delta }_{L0}\right].$$

(16)

Note that in the above, I have augmented [32] (p. 468 No. 7.3.1.207) and [33] with the factor $\left[1+{\delta }_{L0}\right]$ that allows the conversion to be extended downward from their restriction: $L>0$. When multiplied by the equivalent factor in Wimp’s Chebyshev expansion (14), one obtains $\left(2-{\delta }_{L0}\right)\left[1+{\delta }_{L0}\right]\equiv 2$ for all L. This is a strong argument for using the “sums should simply be sums” convention over Clenshaw’s for the present analytical work.

The final step in the proof is to convert each ${}_{2}F_1$ hypergeometric function into a finite sum over ratios of Pochhammer symbols times inverse powers of x. (Let us use m for the summation index). One finds that, of the finite sum in (16) for a given L, the only term that contributes a power $x^{2h}$ is

$$2^{2L-1}{x}^{2L}\sum _{m=L-h}^{L-h}{\displaystyle \frac{x^{-2m}{\left(\frac{1}{2}-L\right)}_m{(-L)}_m}{m!{(1-2L)}_m}}\left[1+{\delta }_{L0}\right],$$

(17)

which may be more compactly written as

$$2^{2L-1}{x}^{2L-2(L-h)}{\displaystyle \frac{{\left(\frac{1}{2}-L\right)}_{L-h}{(-L)}_{L-h}}{(L-h)!{(1-2L)}_{L-h}}}\left[1+{\delta }_{L0}\right].$$

(18)

Noting that multiplying the factor $\left(2-{\delta }_{L0}\right)$ from (14) by the above $\left[1+{\delta }_{L0}\right]$ gives another factor of 2 for all L, which completes the proof of (10). □

To numerically verify the lowest 43 summed series for $k\to 8$, one has to take the upper limit on the number of terms in the series ≥$h+18$ in order to obtain a percent difference between left- and right-hand sides that is ≤${10}^{-33}$, because the first h terms in the series do not contribute. For $k\to 5$, this reduces somewhat to ≥$h+15$. For $h=0$, one needs 24 terms and 20 terms, respectively.

The first 22 terms in the Chebyshev polynomial expansion of $J_1\left(kx\right)$ (12), with $k=1$ and 8, are given in Appendix A.2. The consequent summed series associated with the power $x^{2h+1}$ in the Chebyshev expansion (12) is given in the following theorem:

Theorem 4.

For integer h and for any values of k,

$$\begin{array}{ccc}\hfill \sum _{L=1}^{\infty }& & {\displaystyle \frac{{(-1)}^L{2}^{-2L-1}{\left(\frac{1}{2}-L\right)}_{L-h}{(-L)}_{L-h}}{L!\mathsf{\Gamma }(L+2)(L-h)!{(1-2L)}_{L-h}}}{k}^{2L+1}{}_{1}F_2\left(L+{\displaystyle \frac{1}{2}};L+2,2L+1;-{\displaystyle \frac{k^2}{4}}\right)\hfill \\ & =& {\displaystyle \frac{{(-1)}^h{2}^{-2h-1}}{h!\mathsf{\Gamma }(h+2)}}{k}^{2h+1}.\hfill \end{array}$$

(19)

Proof of Theorem 4.

What changes in the proof as we move from $\nu =0\to 1$ is contained in the three factors

$${\displaystyle \frac{2^{-4L-\nu }}{\mathsf{\Gamma }(L+\nu +1)}}{\left.{}_{1}F_2\left(L+{\displaystyle \frac{1}{2}};L+\nu +1,2L+1;-{\displaystyle \frac{k^2}{4}}\right)\right|}_{\nu \to 1}$$

(20)

in the coefficient $C_{L\nu }\left(k\right)$ of the defining series (12), while nothing does in the Chebyshev polynomial that multiplies it. Thus, nothing changes in the transformations (16)–(18) except that we now associate (17) and (18) with a power $x^{2h}$ multiplied by ${\left(kx\right)}^{\nu }$. Indeed, we have not only proved Theorem 4, but also its extension to a series associated with the power $x^{2h+\nu }$ in the general-$\nu$ case: □

Theorem 5.

For integer h and for any values of k and ν,

$$\begin{array}{ccc}\hfill \sum _{L=1}^{\infty }& & {\displaystyle \frac{{(-1)}^L{2}^{-2L-\nu }{\left(\frac{1}{2}-L\right)}_{L-h}{(-L)}_{L-h}}{L!(L-h)!{(1-2L)}_{L-h}\mathsf{\Gamma }(L+\nu +1)}}{k}^{2L+\nu }{}_{1}F_2\left(L+{\displaystyle \frac{1}{2}};2L+1,L+\nu +1;-{\displaystyle \frac{k^2}{4}}\right)\hfill \\ & =& {\displaystyle \frac{{(-1)}^h{2}^{-2h-1}}{h!\mathsf{\Gamma }(h+2)}}{k}^{2h+1}.\hfill \end{array}$$

(21)

To verify the lowest 43 summed series for $k\to 8$, one generally has to take the upper limit on the number of terms in the series ≥$h+20$ in order to obtain a percent difference between left- and right-hand sides that is ≤${10}^{-33}$, because the first h terms in the series do not contribute. For $k\to 5$, this reduces somewhat to ≥$h+16$. For $h=0$, one needs 23 terms and 20 terms, respectively.

For large indices, such as $\nu =17$, for instance, with $h=5$, $k=8$, the two sides of (21) sides diverge after 45 digits: $-1.335586213327781269795862205505422996145960793\times {10}^{-7}$. For small values of $\nu$, however, neither side gives an accuracy beyond the 13th post-decimal place in the computer algebra program Mathematica 7 despite a command to do so, giving $-68.7857424612620$ for $h=5$, $k=8$, and $\nu =0.17$. Complex values of $\nu$ likewise gave only 14 decimal placers in Mathematica, such as $1.15092097688009+1.83846320788943i$ for $h=5$, $k=8$, and $\nu =17+30.3i$. Mathematica 13 likewise gives this more limited, but still excellent, accuracy.

In the prior paper [27], we noted that because the modified Bessel functions of the first kind $I_N\left(kx\right)$ are related to the ordinary Bessel functions by the relation [31] (p. 961 No. 8.406.3),

$$I_n\left(z\right)=i^{-n}{J}_n\left(iz\right),$$

(22)

one merely needs to multiply by $i^{-n}$ and set $k=i$ in (2) to obtain the $I_0\left(x\right)$ Fourier–Legendre series. Furthermore, one sees that $I_0$ expressed in powers of x is simply the $J_0$ version with all of the negative signs reversed. This is not true of (1) because the arguments of the Legendre polynomials do not undergo $x\to ix$ since they derive from the definition of the Fourier–Legendre series (1). The k-dependence is entirely within the coefficients $a_{LN}\left(k\right)$.

Therefore, the $I_0$ Legendre series expansion leads to no new set of summed series since these would simply be (7) with $k=i\kappa$. This is also the case for a Chebyshev expansion. Clenshaw [34] confirms this for $h=0$ on pp. 34–35.

3. Summed Series Involving ${}_{\mathbf{1}}\mathit{F}_{\mathbf{2}}$ Hypergeometric Functions from Gegenbauer Polynomial Expansions of Bessel Functions

We wish to prove the following theorem for summed series derived from Gegenbauer polynomial expansions of the $J_0\left(kx\right)$ Bessel function:

Theorem 6.

For integer h and for any values of k and λ,

$$\begin{array}{ccc}\hfill \sum _{L=0}^{\infty }& & {\displaystyle \frac{{(-2)}^{2L}{(-L)}_h{\left(\lambda +\frac{1}{2}\right)}_{2L}{(L+\lambda )}_h}{\sqrt{\pi }h!{\left(\frac{1}{2}\right)}_h{\left(L+\frac{1}{2}\right)}_{\frac{1}{2}}(L+\lambda ){\left(2\lambda \right)}_{2L}{(2L+2\lambda )}_{2L}B(\lambda ,L+1)}}{k}^{2L}{}_{1}F_2\left(L+{\displaystyle \frac{1}{2}};L+1,2L+\lambda +1;-{\displaystyle \frac{k^2}{4}}\right)\hfill \\ & ={\displaystyle \frac{{(-1)}^h{2}^{-2h}{k}^{2h}}{h!\mathsf{\Gamma }(h+1)}}& .\hfill \end{array}$$

(23)

Proof of Theorem 6.

Although he does not explicitly do so, one may use Wimp’s Jacobi expansion [30] to find Gegenbauer polynomial expansions of Bessel functions, since [31] (p. 1061 No. 8.962.4)

$$P_{2n}^{\left(\lambda -{\displaystyle \frac{1}{2}},\lambda -{\displaystyle \frac{1}{2}}\right)}\left(z\right)={\displaystyle \frac{{\left(\lambda +\frac{1}{2}\right)}_{2n}{C}_{2n}^{\lambda }\left(z\right)}{{\left(2\lambda \right)}_{2n}}}.$$

(24)

Like those in the second section, the following expansion applies to both integer and non-integer indices:

$$J_{\nu }\left(kx\right)={\left(kx\right)}^{\nu }\sum _{L=0}^{\infty }{b}_{L\nu }\left(k\right)C_{2L}^{\lambda }\left(x\right),-1\le x\le 1$$

(25)

where the coefficients are given by the orthogonality of the Chebyshev polynomials,

$$b_{L\nu }\left(k\right)={\displaystyle \frac{2^{2\lambda -1}\left(2L\right)!(2L+\lambda )\mathsf{\Gamma }{\left(\lambda \right)}^2}{\pi \mathsf{\Gamma }(2L+2\lambda )}}{\int }_{-1}^1\left(J_{\nu }\left(kx\right){\left(kx\right)}^{-\nu }\right){\left(1-x^2\right)}^{-{\displaystyle \frac{1}{2}}+\lambda }{C}_{2L}^{\lambda }\left(x\right)dx,$$

(26)

as

$$\begin{array}{ccc}{b}_{L\nu }\left(k\right)& ={\displaystyle \frac{{(-1)}^L{a}^{2L}{2}^{2L-\nu }{\left(\lambda +\frac{1}{2}\right)}_{2L}}{\sqrt{\pi }{\left(2\lambda \right)}_{2L}{(2L+2\lambda )}_{2L}{\left(L+\frac{1}{2}\right)}_{\nu +\frac{1}{2}}}}& {}_{1}F_2\left(L+{\displaystyle \frac{1}{2}};2L+\lambda +1,L+\nu +1;-{\displaystyle \frac{a^2}{4}}\right)\end{array}.$$

(27)

The first 22 terms in the Gegenbauer polynomial expansion of $J_0\left(kx\right)$ (25) are given in Appendix A.3, with $k=1$ and arbitrarily taking $\lambda ={\displaystyle \frac{1}{4}}$. One could, of course, test the technique using any value of $\lambda$, but since $T_{\nu }\left(z\right)={\displaystyle \frac{1}{2}}\nu C_{\nu }^0\left(z\right)$ and $P_{\nu }\left(z\right)=C_{\nu }^{{\displaystyle \frac{1}{2}}}\left(z\right)$, the choice $\lambda ={\displaystyle \frac{1}{4}}$ seemed like the next most interesting value.

To extract the powers, we use the conversion for Gegenbauer polynomials that is equivalent to (16), which is [31] (GR5 p. 1051 No. 8.932.2)

$$C_{2L}^{\lambda }\left(x\right)={\displaystyle \frac{{(-1)}^L}{(L+\lambda )B(\lambda ,L+1)}}{}_{2}F_1\left(-L,L+\lambda ;{\displaystyle \frac{1}{2}};x^2\right).$$

(28)

The final step in the proof is to convert each ${}_{2}F_1$ hypergeometric function into a finite sum over ratios of Pochhammer symbols times powers of x. (Let us use m for the summation index). We find that, of the finite sum in (28) for a given L, the only term that contributes a power $x^{2h}$ is

$${\displaystyle \frac{{(-1)}^L}{(L+\lambda )B(\lambda ,L+1)}}\sum _{m=h}^h{\displaystyle \frac{x^{2m}{(-L)}_m{(L+\lambda )}_m}{m!{\left(\frac{1}{2}\right)}_m}},$$

(29)

which may be more compactly written as

$${\displaystyle \frac{{(-1)}^L}{(L+\lambda )B(\lambda ,L+1)}}{\displaystyle \frac{x^{2h}{(-L)}_h{(L+\lambda )}_h}{h!{\left(\frac{1}{2}\right)}_h}},$$

(30)

which completes the proof of (23). □

To verify the lowest 43 summed series for $k\to 8$, one generally has to take the upper limit on the number of terms in the series ≥$h+20$ in order to obtain a percent difference between left- and right-hand sides that is ≤${10}^{-33}$, because the first h terms in the series do not contribute. For $k\to 5$, this reduces somewhat to ≥$h+16$. For $h=0$, one needs 23 terms and 20 terms, respectively.

This theorem has an identical right-hand side as for the Legendre (7) and Chebyshev (10) versions, and it holds for every value of $\lambda$. That is, we have just summed an infinite set of infinite sets of infinite series involving ${}_{1}F_2$ hypergeometric functions. To see how this plays out in practice, consider two extreme values, $\lambda =2^{\pm 20}$. For $h=1$ (associated with $x^2$) and $\lambda =2^{-20}$, the first eight terms sum as

$$\begin{array}{c}0\\ -0.234776027081720679198861338236978\\ -0.0149856953860168611951004494702182\\ -0.000236617512932378657466894715711978\\ -1.653516950294282858347187372908306\times {10}^{-6}.\\ -6.486087810927263603219843086719685\times {10}^{-9}\\ -1.62636408715893081661576487444697\times {10}^{-11}\\ -2.82986734360173162046207736379133\times {10}^{-14}-\cdots \\ =-0.24999999999999996\end{array}$$

(31)

For $\lambda =2^{20}$, the second term is almost sufficient by itself:

$$\begin{array}{c}0\\ -0.249999955296648756645694568085746\\ -4.470334680250094078684477109090418\times {10}^{-8}\\ -4.44085305583632491122417076683932\times {10}^{-14}\\ -3.08808728006191746252089269280760\times {10}^{-22}.\\ -1.65656014384509551797626946530621\times {10}^{-29}\\ -7.24073285415562941500526940820453\times {10}^{-37}\\ -2.67164795422661275973040680548870\times {10}^{-44}-\cdots \\ =0.250000000000000000000000000000000\end{array}$$

(32)

The final theorem we wish to prove is for series associated with the power $x^{2h+\nu }$ derived from the Gegenbauer polynomial expansion of $J_{\nu }\left(kx\right)$, the general $\nu$ case, which may be written as follows:

Theorem 7.

For integer h and for any values of k, λ, and ν,

$$\begin{array}{ccc}\hfill \sum _{L=0}^{\infty }& & {\displaystyle \frac{{(-1)}^{2L}{2}^{2L-\nu }{(-L)}_h{\left(\lambda +\frac{1}{2}\right)}_{2L}{(L+\lambda )}_h}{\sqrt{\pi }h!{\left(\frac{1}{2}\right)}_h(L+\lambda ){\left(2\lambda \right)}_{2L}{(2L+2\lambda )}_{2L}{\left(L+\frac{1}{2}\right)}_{\nu +\frac{1}{2}}B(\lambda ,L+1)}}{k}^{2L+\nu }\hfill \\ & \times & {}_{1}F_2\left(L+{\displaystyle \frac{1}{2}};2L+\lambda +1,L+\nu +1;-{\displaystyle \frac{k^2}{4}}\right)\hfill \\ & =& {\displaystyle \frac{{(-1)}^h{2}^{-2h-\nu }{k}^{2h+\nu }}{h!\mathsf{\Gamma }(h+\nu +1)}}.\hfill \end{array}$$

(33)

Proof of Theorem 7.

One may build the general case by examining what must be done to sum the series associated with the power $x^{2h+1}$ derived from the Gegenbauer polynomial expansion of $J_1\left(kx\right)$, whose 22-term Gegenbauer polynomial expansion we display in Appendix A.3, with $k=1$ and again arbitrarily taking $\lambda ={\displaystyle \frac{1}{4}}$.

What changes in the proof as we move from $\nu =0\to 1$ is contained in the three factors

$${\displaystyle \frac{2^{2L-\nu }}{{\left(L+\frac{1}{2}\right)}_{\nu +\frac{1}{2}}}}{\left.{}_{1}F_2\left(L+{\displaystyle \frac{1}{2}};2L+\lambda +1,L+\nu +1;-{\displaystyle \frac{k^2}{4}}\right)\right|}_{\nu \to 1}$$

(34)

in the coefficient $b_{L\nu }\left(k\right)$ of the defining series (25), while nothing does in the Gegenbauer polynomial that multiplies it. Thus, nothing changes in the transformations (28)–(30) except that we now associate (29) and (30) with a power $x^{2h}$ multiplied by ${\left(kx\right)}^{\nu }$. This completes the proof of the summed series associated with the power $x^{2h+\nu }$ in the general-$\nu$ case. □

To verify the lowest 43 summed series for $k\to 8$, one generally has to take the upper limit on the number of terms in the series ≥$h+20$ in order to obtain a percent difference between left- and right-hand sides that is ≤${10}^{-33}$, because the first h terms in the series do not contribute. For $k\to 5$, this reduces somewhat to ≥$h+16$. For $h=0$, one needs 23 terms and 20 terms, respectively.

The summed series derived from the Gegenbauer polynomial expansions of $J_{\nu }\left(x\right)$ may be found for any value of $\nu$, not just integer values, given that it is derived from Wimp’s Jacobi expansion [30]. Thus, we have just summed an infinite set of infinite sets of doubly infinite series involving ${}_{1}F_2$ hypergeometric functions since the expression holds for every value of $\lambda$ and holds for every value of $\nu$.

Since $T_{\nu }\left(z\right)={\displaystyle \frac{1}{2}}\nu C_{\nu }^0\left(z\right)$, by setting $\lambda =0$ in (33) one may obtain a modest variation on the form given in (21), since we here use a ${}_{2}F_1$ hypergeometric function whose argument is $x^2$ in (33) and used a ${}_{2}F_1$ hypergeometric function whose argument is $x^{-2}$ to prove (21).

An extension of the Legendre sets (7) and (9) to larger integer values of $\nu$ is not obvious, but one can obtain such a form directly from (33) since $P_{\nu }\left(z\right)=C_{\nu }^{{\displaystyle \frac{1}{2}}}\left(z\right)$, which applies even for non-integer values of $\nu$.

4. Conclusions

I have shown how to sum doubly infinite sets of infinite series involving ${}_{1}F_2$ hypergeometric functions, derived from Chebyshev polynomial expansions of Bessel functions of the first kind $J_{\nu }\left(kx\right)$, and the trebly infinite sets of infinite series involving ${}_{1}F_2$ hypergeometric functions from the Gegenbauer polynomial expansions of $J_{\nu }\left(kx\right)$. The utility of any one of these summed series for future researchers is, of course, not guaranteed, but given the relative paucity of infinite series whose values are known (e.g., 24 pages in Gradshteyn and Ryzhik compared to their 900 pages of known integrals), one hopes that adding such multiply-infinite sets of infinite series of ${}_{1}F_2$ functions whose values are now known will be of use to some.