Certain Summation and Operational Formulas Involving Gould–Hopper–Lambda Polynomials

Abstract

This manuscript introduces the family of Gould–Hopper–Lambda polynomials and establishes their quasi-monomial properties through the umbral method. This approach serves as a powerful mechanism to analyze the characteristic of multi-variable special polynomials. Several summation formulas for these polynomials are explored, and their operational identities are obtained using partial differential equations. The corresponding results for Hermite–Lambda polynomials are also obtained. In addition, a conclusion is given.

1. Introduction

In recent years, special multi-variable functions in mathematical physics have experienced considerable development. Notably, special polynomials in three variables provide innovative analytical techniques for addressing a variety of partial differential equations frequently seen in physical contexts. Many of these special functions and their generalizations are derived from real-world physical problems. Several researchers have explored three versions of the Hermite, Laguerre, and Appell polynomials, along with their convolutions (see, for example, [1,2,3,4,5]).

Recently, Dattoli and his collaborators have investigated a novel family of two-variable special polynomials [6]. Their work introduces a new class of polynomials called Lambda polynomials ${\lambda }_n(\xi ,\rho )$, which serve as a connection between Laguerre and trigonometric functions [6]. The umbral technique can be viewed as a subset of symbolic methods, as both aim to derive an operator for special polynomials that allows for their expression in a binomial form, though their approaches differ. In the umbral technique, suitable umbra are developed based on specific conditions of the special functions. This leads to a connection between the functions and the corresponding binomial expressions they satisfy, all within the framework of umbral formalism (see, for example, [7,8,9]). Recently, the properties of $\lambda$–Bessel functions have been explored by Zainab et al. [10].

The series representation and umbral image of the Lambda polynomials are presented as follows:

$${\lambda }_n(\xi ,\rho )=n!\sum _{r=0}^n{\displaystyle \frac{{(-1)}^r}{\left(2r\right)!(n-r)!}}{\rho }^{n-r}{\xi }^r,\left|\xi \right|<\infty ,\left|\rho \right|<\infty ,$$

(1)

and

$${\lambda }_n(\xi ,\rho )=(\rho -\widehat{C}\xi ){\psi }_0,$$

(2)

respectively, where $\widehat{C}$ denotes an umbral operator introduced by Dattoli et al. [6], which operates on the vacuum polynomial or function ${\psi }_0$ as:

$${\widehat{C}}^r{\psi }_0={\displaystyle \frac{\Gamma (r+1)}{\Gamma (2r+1)}}(r\in \mathbb{R}),$$

(3)

such that

$${\widehat{C}}^n{\widehat{C}}^m={\widehat{C}}^{n+m}.$$

(4)

The term vacuum polynomial is borrowed from physics, on which the operator works to generate the polynomials.

The exponential and ordinary generating functions of Lambda polynomials [6] are represented as:

$$\sum _{n=0}^{\infty }{\lambda }_n(\xi ,\rho ){\displaystyle \frac{t^n}{n!}}=e^{\rho t}cos\left(\sqrt{\xi t}\right),\left|\xi \right|<\infty ,\left|\rho \right|<\infty ,$$

(5)

and

$$\sum _{n=0}^{\infty }{\lambda }_n(\xi ,\rho )t^n={\displaystyle \frac{1}{1-\rho t}}{e}_0\left({\displaystyle \frac{\xi t}{1-\rho t}}\right),$$

(6)

respectively, where the cosine function and $e_0\left(\xi \right)$ functions are defined as:

$$cos\left(\xi \right)=e^{-\widehat{C}{\xi }^2}{\psi }_0,$$

(7)

$$e_0\left(\xi \right)=\sum _{r=0}^{\infty }{(-1)}^r{\displaystyle \frac{r!}{\left(2r\right)!}}{\xi }^r,$$

(8)

respectively.

To address problems across various fields arising from mathematics, ranging from the topic of partial differential equations to the group theory of multi-indices and multi-variables, special functions have become essential. The original theory of multi-variable and multi-index Hermite polynomials was originally given by Hermite [11]. Hermite polynomials appear as a special case of Appell polynomials in the field of combinatorics that follow umbral calculus in the numerical method and analysis through Gaussian quadrature, and also in physical mathematics, where they describe the eigenstates of the quantum harmonic oscillator and serve as solutions for the Schrödinger equation [12].

The Gould–Hopper polynomials are viewed as an extension of the two-variable Hermite–Kampé de Fériet polynomials (2VHKdFP) $H_n^{\left(m\right)}(\xi ,\rho )$. These polynomials, denoted as (GHPs) $H_n^{\left(m\right)}(\xi ,\rho )$ are defined through the following generating function and series representation for any positive integer m [13]:

$$\sum _{n=0}^{\infty }{H}_n^{\left(m\right)}(\xi ,\rho ){\displaystyle \frac{t^n}{n!}}=e^{\xi t+\rho t^m},\left|\xi \right|<\infty ,\left|\rho \right|<\infty ,$$

(9)

and

$$H_n^{\left(m\right)}(\xi ,\rho )=n!\sum _{r=0}^{\left[{\displaystyle \frac{n}{m}}\right]}{\displaystyle \frac{{\xi }^{n-mr}{\rho }^r}{(n-mr)!r!}},$$

(10)

respectively.

The GHPs $H_n^{\left(m\right)}(\xi ,\rho )$ serve as solutions to the extended heat equation [14]:

$${\displaystyle \frac{\partial }{\partial \rho }}f(\xi ,\rho )={\displaystyle \frac{{\partial }^m}{\partial {\xi }^m}}f(\xi ,\rho ),$$

(11)

where the initial condition is $f(\xi ,0)={\xi }^n.$

The GHPs $H_n^{\left(m\right)}(\xi ,\rho )$ are defined by the given operational rule: [14]:

$$H_n^{\left(m\right)}(\xi ,\rho )=exp\left(\rho D_{\xi }^m\right)\left\{{\xi }^n\right\},$$

(12)

where

$$D_{\xi }:={\displaystyle \frac{\partial }{\partial \xi }}.$$

(13)

Gould and Hopper [13] originally denoted the Gould–Hopper polynomials as $g_n^s(\xi ,\rho )$. However, because of their direct connection to Hermite polynomials, we adopt the notation $H_n^{\left(m\right)}(\xi ,\rho )$ in this paper, as used by other researchers.

We should note that for $m=2$,

$$H_n^{\left(2\right)}(\xi ,\rho )=H_n(\xi ,\rho ).$$

(14)

Here, $H_n(\xi ,\rho )$ represents the 2VHKdFP, which are represented by given generating function [15]:

$$\sum _{n=0}^{\infty }{H}_n(\xi ,\rho ){\displaystyle \frac{t^n}{n!}}=e^{\xi t+\rho t^2}.$$

(15)

Dattoli et al. obtained the umbral image of $H_n(\xi ,\rho )$ as [16]:

$$H_n(\xi ,\rho )={(\xi +{}_{\sqrt{\rho }}\widehat{h})}^n{\varphi }_0,$$

(16)

where ${\varphi }_0$ is treated as a polynomial vacuum and ${}_{\sqrt{\rho }}\widehat{h}$ is an umbra acting on the polynomial vacuum ${\varphi }_0$ to generate the 2VHP $H_n(\xi ,\rho )$ given by

$${{}_{\sqrt{\rho }}\widehat{h}}^r{\varphi }_0={\displaystyle \frac{{\rho }^{\frac{r}{2}}r!}{\Gamma (\frac{r}{2}+1)}}\left|cosr{\displaystyle \frac{\pi }{2}}\right|({\varphi }_0\ne 0),$$

(17)

the exponential of which is given by

$$e^{{}_{\sqrt{\rho }}\widehat{h}t}{\varphi }_0=e^{\rho t^2}.$$

(18)

Dattoli et al. gave the following binomial [16,17] expression for GHPs $H_n^{\left(m\right)}(\xi ,\rho )$:

$$H_n^{\left(m\right)}(\xi ,\rho )={(\xi +{\widehat{h}}_m\sqrt[m]{\rho })}^n{\varphi }_0,$$

(19)

by introducing the umbra [16]

$${\widehat{h}}_m^r{\varphi }_0={\displaystyle \frac{r!}{\Gamma \left(\frac{r}{m}+1\right)}}{\delta }_{m⌊{\displaystyle \frac{r}{m}}⌋,r},$$

(20)

in which ${\widehat{h}}_m$ is the generalization of $\widehat{h}$ for the case $m=2$.

The exponential value of ${\widehat{h}}_m$ is given by

$$e^{{\widehat{h}}_m\sqrt[m]{\rho }t}{\varphi }_0=e^{\rho t^m}.$$

(21)

The concept of poweroid abstraction, introduced by Steffensen [18], provides the foundation for the notion of monomiality. This concept was further refined and established by Dattoli [19].

The monomiality principle states that:

The term “quasi-monomial” describes the polynomial sequence ${\left\{s_n\left(\xi \right)\right\}}_{n=0}^{\infty }$, which has two operators $\widehat{M}$ and $\widehat{P}$, namely multiplicative operator and derivative operator, respectively, satisfying the given relations [19]:

$$\widehat{M}\left\{s_n\left(\xi \right)\right\}=s_{n+1}\left(\xi \right),$$

(22)

and

$$\widehat{P}\left\{s_n\left(\xi \right)\right\}=n{s}_{n-1}\left(\xi \right).$$

(23)

The following commutation relation satisfied by $\widehat{M}$ and $\widehat{P}$:

$$[\widehat{P},\widehat{M}]=\widehat{P}\widehat{M}-\widehat{M}\widehat{P}=\widehat{1}.$$

(24)

Hence, $\widehat{M}$ and $\widehat{P}$ operators exhibit a Weyl group structure [19]. By utilizing these two operators, various properties of the polynomial $s_n\left(\xi \right)$ can be derived. If differential realizations are satisfied by $\widehat{M}$ and $\widehat{P}$ operators, then the polynomials $s_n\left(\xi \right)$ satisfy the following differential equation:

$$\widehat{M}\widehat{P}\left\{s_n\left(\xi \right)\right\}=n{s}_n\left(\xi \right).$$

(25)

In view of Equation (22), the $s_n\left(\xi \right)$ can be explicitly formed as:

$$s_n\left(\xi \right)={\widehat{M}}^n\left\{s_0\left(\xi \right)\right\}.$$

(26)

Section 2 deals with the introduction of the Gould–Hopper–Lambda polynomials (GHLPs) ${}_H\lambda _n^{\left(m\right)}(\xi ,\rho ,\eta )$ and then investigates several characteristics of this family, such as generating function, quasi-monomiality, and differential equations. In Section 3 and Section 4, summation formulas, partial differential equations, and some operational identities of GHLPs ${}_H\lambda _n^{\left(m\right)}(\xi ,\rho ,\eta )$ using umbral–algebraic approach are established.

2. Gould–Hopper–Lambda Polynomials

In this section, first we introduce Gould–Hopper–Lambda polynomials (GHLPs) ${}_H\lambda _n^{\left(m\right)}(\xi ,\rho ,\eta )$ using the umbral method. To do this, we convolve Gould–Hopper polynomials with Lambda polynomials using the umbral images of both polynomials.

By replacing $\rho$ in Equation (2) with the umbral image of the Gould–Hopper polynomials $H_n^{\left(m\right)}(\xi ,\rho )$ as given in Equation (19), we obtain the following umbral image for the GHLPs:

$${}_H\lambda _n^{\left(m\right)}(\xi ,\rho ,\eta )={(\xi +{\widehat{h}}_m\sqrt[m]{\rho }-\widehat{C}\eta )}^n{\varphi }_0{\psi }_0.$$

(27)

We obtain the following generating function for GHLPs ${}_H\lambda _n^{\left(m\right)}(\xi ,\rho ,\eta )$:

Theorem 1. 

The given generating function is satisfied by Gould–Hopper–Lambda polynomials ${}_H\lambda _n^{\left(m\right)}(\xi ,\rho ,\eta )$

$$\sum _{n=0}^{\infty }{}_H\lambda _n^{\left(m\right)}(\xi ,\rho ,\eta ){\displaystyle \frac{t^n}{n!}}=e^{\xi t+\rho t^m}cos\left(\sqrt{\eta t}\right).$$

(28)

Proof. 

Using Weyl identity given in [20]:

$$e^{\widehat{A}+\widehat{B}}=e^{\widehat{A}}{e}^{\widehat{B}}{e}^{{\displaystyle \frac{-k}{2}}},k=[\widehat{A},\widehat{B}](k\in \mathbb{C}),$$

(29)

in Equation (27), we obtain

$$\sum _{n=0}^{\infty }{}_H\lambda _n^{\left(m\right)}(\xi ,\rho ,\eta ){\displaystyle \frac{t^n}{n!}}=exp\left((\xi +{\widehat{h}}_m\sqrt[m]{\rho })t\right)exp(-\widehat{C}\eta t){\varphi }_0{\psi }_0,$$

(30)

which, upon simplifying by using Equation (7), gives

$$\sum _{n=0}^{\infty }{}_H\lambda _n^{\left(m\right)}(\xi ,\rho ,\eta ){\displaystyle \frac{t^n}{n!}}=e^{\xi t+\rho t^m}{e}^{(-\widehat{C}\eta t)}{\varphi }_0{\psi }_0.$$

(31)

In the above equation, using Equation (21) on the right-hand side, we obtain assertion (28). □

Remark 1. 

For $m=2$ in Theorem 1, we obtain the following generating function for the Hermite–Lambda polynomials ${}_{\lambda }H_n(\xi ,\rho ,\eta )$:

$$\sum _{n=0}^{\infty }{}_{\lambda }H_n(\xi ,\rho ,\eta ){\displaystyle \frac{t^n}{n!}}=e^{\xi t+\rho t^2}cos\left(\sqrt{\eta t}\right).$$

(32)

We yield the following series expansion form of GHLPs ${}_H\lambda _n^{\left(m\right)}(\xi ,\rho ,\eta )$:

Theorem 2. 

The given series definition is satisfied by Gould–Hopper–Lambda polynomials ${}_H\lambda _n^{\left(m\right)}(\xi ,\rho ,\eta )$

$${}_H\lambda _n^{\left(m\right)}(\xi ,\rho ,\eta )=\sum _{r=0}^n\left({\displaystyle \genfrac{}{}{0pt}{}{n}{r}}\right)H_{n-r}^{\left(m\right)}(\xi ,\rho ){(-1)}^r{\eta }^r{\displaystyle \frac{\Gamma (r+1)}{\Gamma (2r+1)}}.$$

(33)

Proof. 

Binomially expanding the right-hand side, present in Equation (27), we yield

$${}_H\lambda _n^{\left(m\right)}(\xi ,\rho ,\eta )=\sum _{r=0}^n\left({\displaystyle \genfrac{}{}{0pt}{}{n}{r}}\right){(\xi +{\widehat{h}}_m\sqrt[m]{\rho })}^{n-r}{(-\widehat{C}\eta )}^r{\varphi }_0{\psi }_0.$$

(34)

In the above equation, using Equations (3) and (19) on the right-hand side, we obtain assertion (33). □

Remark 2. 

For $m=2$ in Theorem 2, we obtain the following series expansion for Hermite–Lambda polynomials ${}_{\lambda }H_n(\xi ,\rho ,\eta )$:

$${}_{\lambda }H_n(\xi ,\rho ,\eta )=\sum _{r=0}^n\left({\displaystyle \genfrac{}{}{0pt}{}{n}{r}}\right)H_{n-r}(\xi ,\rho ){(-1)}^r{\eta }^r{\displaystyle \frac{\Gamma (r+1)}{\Gamma (2r+1)}}.$$

(35)

We obtain the following quasi-monomiality principle of GHLPs ${}_H\lambda _n^{\left(m\right)}(\xi ,\rho ,\eta )$:

Theorem 3. 

The Gould–Hopper–Lambda polynomials ${}_H\lambda _n^{\left(m\right)}(\xi ,\rho ,\eta )$ satisfy the quasi-monomiality principle with the following multiplicative and derivative operators:

$${\widehat{M}}_{G\lambda }=\xi +{\widehat{h}}_m\sqrt[m]{\rho }-\widehat{C}\eta ,$$

(36)

and

$$P_{G\lambda }=D_{\xi },$$

(37)

respectively.

Proof. 

Operating $(\xi +{\widehat{h}}_m\sqrt[m]{\rho }-\widehat{C}\eta )$ on both the sides of Equation (27) and then by making use of Equation (27) on the right-hand side of the resultant equation, we yield

$$(\xi +{\widehat{h}}_m\sqrt[m]{\rho }-\widehat{C}\eta ){}_H\lambda _n^{\left(m\right)}(\xi ,\rho ,\eta )={}_H\lambda _{n+1}^{\left(m\right)}(\xi ,\rho ,\eta ),$$

(38)

which, upon comparison with Equations (22) and (38), yields assertion (36).

Again, differentiating Equation (27) with respect to ξ and then using Equation (27) in the resulting equation, we obtain

$${\displaystyle \frac{\partial }{\partial \xi }}{}_H\lambda _n^{\left(m\right)}(\xi ,\rho ,\eta )=n{}_H\lambda _{n+1}^{\left(m\right)}(\xi ,\rho ,\eta ),$$

(39)

which, upon comparison with Equations (23) and (39), yields assertion (37). □

Remark 3. 

For $m=2$ in Theorem 3, we obtain the following multiplicative and derivative operators for Hermite–Lambda polynomials ${}_{\lambda }H_n(\xi ,\rho ,\eta )$:

$${\widehat{M}}_{H\lambda }=\xi +{}_{\sqrt{\rho }}\widehat{h}-\widehat{C}\eta ,$$

(40)

and

$$P_{H\lambda }=D_{\xi },$$

(41)

respectively.

Using Equations (25), (36) and (37), we can prove the following theorem for differential equation of GHLPs ${}_H\lambda _n^{\left(m\right)}(\xi ,\rho ,\eta )$:

Theorem 4. 

The umbral differential equation satisfied by the Gould–Hopper–Lambda polynomials ${}_H\lambda _n^{\left(m\right)}(\xi ,\rho ,\eta )$ is given by

$$\left(\left(\xi +{\widehat{h}}_m\sqrt[m]{\rho }-\widehat{C}\eta )\right){\displaystyle \frac{\partial }{\partial \xi }}-n\right){}_H\lambda _n^{\left(m\right)}(\xi ,\rho ,\eta )=0.$$

(42)

Remark 4. 

For $m=2$ in Theorem 4, we obtain the following differential equation for Hermite–Lambda polynomials ${}_{\lambda }H_n(\xi ,\rho ,\eta )$:

$$\left(\left(\xi +{}_{\sqrt{\rho }}\widehat{h}-\widehat{C}\eta )\right){\displaystyle \frac{\partial }{\partial \xi }}-n\right){}_{\lambda }H_n(\xi ,\rho ,\eta )=0.$$

(43)

3. Summation Formulas

This section contains several summation formulas for the Gould–Hopper–Lambda polynomials ${}_H\lambda _n^{\left(m\right)}(\xi ,\rho ,\eta )$ using the series re-arrangement property.

Now, we yield the following summation formula for GHLPs ${}_H\lambda _n^{\left(m\right)}(\xi ,\rho ,\eta )$:

Theorem 5. 

The given summation formula for the Gould–Hopper–Lambda polynomials ${}_H\lambda _n^{\left(m\right)}(\xi ,\rho ,\eta )$ is valid

$${}_H\lambda _n^{\left(m\right)}(\xi +z,\rho ,\eta )=\sum _{k=0}^n\left({\displaystyle \genfrac{}{}{0pt}{}{n}{k}}\right)z^k{}_{\lambda }H_{n-k}^{\left(m\right)}(\xi ,\rho ,\eta ).$$

(44)

Proof. 

Upon replacement of ξ with $\xi +z$ in the Equation (28), we find

$$\begin{array}{cc}\hfill \sum _{n=0}^{\infty }{}_H\lambda _n^{\left(m\right)}(\xi +z,\rho ,\eta ){\displaystyle \frac{t^n}{n!}}=& exp\left((\xi +z)t+\rho t^m\right)cos\left(\sqrt{\eta t}\right)\hfill \\ \hfill =& exp(\xi t+\rho t^m)exp\left(zt\right)cos\left(\sqrt{\eta t}\right).\hfill \end{array}$$

(45)

Using Equation (28) after expanding the second exponential of the right-hand side of the above equation binomially, we yield

$$\sum _{n=0}^{\infty }{}_H\lambda _n^{\left(m\right)}(\xi +z,\rho ,\eta ){\displaystyle \frac{t^n}{n!}}=\sum _{n=0}^{\infty }\sum _{r=0}^{\infty }{}_H\lambda _n^{\left(m\right)}(\xi ,\eta ;\alpha ,\beta ){\displaystyle \frac{t^n}{n!}}{\displaystyle \frac{z^r{t}^r}{r!}},$$

by making use of series re-arrangement summation formula, it becomes

$$\sum _{n=0}^{\infty }{}_H\lambda _n^{\left(m\right)}(\xi +z,\rho ,\eta ){\displaystyle \frac{t^n}{n!}}=\sum _{n=0}^{\infty }\left(\sum _{r=0}^n\left({\displaystyle \genfrac{}{}{0pt}{}{n}{r}}\right)z^r{}_H\lambda _{n-r}^{\left(m\right)}(\xi ,\rho ,\eta )\right){\displaystyle \frac{t^n}{n!}}.$$

(46)

Comparing the coefficients of equal powers of t from both the sides of Equation (46), we obtain assertion (44). □

Remark 5. 

For $m=2$ in Theorem 5, we obtain the following summation formula for Hermite–Lambda polynomials ${}_{\lambda }H_n(\xi ,\rho ,\eta )$:

$${}_{\lambda }H_n(\xi +z,\rho ,\eta )=\sum _{r=0}^n\left({\displaystyle \genfrac{}{}{0pt}{}{n}{r}}\right)z^r{}_{\lambda }H_{n-r}(\xi ,\rho ,\eta ).$$

(47)

We obtain the following another summation formula of GHLPs ${}_H\lambda _n^{\left(m\right)}(\xi ,\rho ,\eta )$:

Theorem 6. 

The given summation formula for the Gould–Hopper–Lambda polynomials ${}_H\lambda _n^{\left(m\right)}(\xi ,\rho ,\eta )$ is valid

$${}_{\lambda }H_{k+l}^{\left(m\right)}(w,\rho ,\eta )=\sum _{p,q=0}^{k,l}\left({\displaystyle \genfrac{}{}{0pt}{}{k}{p}}\right)\left({\displaystyle \genfrac{}{}{0pt}{}{l}{q}}\right){(w-\xi )}^{p+q}{}_{\lambda }H_{k+l-p-q}^{\left(m\right)}(\xi ,\rho ,\eta ),$$

(48)

where ${\sum }_{p,q=0}^{k,l}:={\sum }_{p=0}^k{\sum }_{q=0}^l.$

Proof. 

Upon replacement of t with $\omega +t$ in (28) and then making use of the series formula [21],

$$\sum _{n=0}^{\infty }f\left(n\right){\displaystyle \frac{{(\xi +\rho )}^n}{n!}}=\sum _{n,r=0}^{\infty }f(n+r){\displaystyle \frac{{\xi }^n}{n!}}{\displaystyle \frac{{\rho }^r}{r!}},$$

(49)

in the resultant equation, we yield the following generating function for GHLPs ${}_H\lambda _n^{\left(m\right)}(\xi ,\rho ,\eta )$:

$$\sum _{k,l=0}^{\infty }{\displaystyle \frac{t^k{\omega }^l}{k!l!}}{}_{\lambda }H_{k+l}^{\left(m\right)}(\xi ,\rho ,\eta )=exp(\xi (t+\omega )+\rho {(t+\omega )}^m)cos\left(\sqrt{\eta (t+u)}\right),$$

which becomes

$$exp(-\xi (t+\omega ))\sum _{k,l=0}^{\infty }{\displaystyle \frac{t^k{\omega }^l}{k!l!}}{}_{\lambda }H_{k+l}^{\left(m\right)}(\xi ,\rho ,\eta )=exp\left(\rho {(t+\omega )}^m\right)cos\left(\sqrt{\eta (t+\omega )}\right).$$

By doing multiplication on both the sides of above equation with $exp\left(w(t+\omega )\right)$ and then, by applying Equation (49) to the left-hand side of the resulting equation, we obtain

$$exp\left(\right(w-\xi \left)\right(t+\omega \left)\right)\sum _{k,l=0}^{\infty }{\displaystyle \frac{t^k{\omega }^l}{k!l!}}{}_{\lambda }H_{k+l}^{\left(m\right)}(\xi ,\rho ,\eta )=\sum _{k,l=0}^{\infty }{\displaystyle \frac{t^k{\omega }^l}{k!l!}}{}_{\lambda }H_{k+l}^{\left(m\right)}(w,\rho ,\eta ),$$

or equivalently

$$\sum _{k,l=0}^{\infty }{\displaystyle \frac{t^k{\omega }^l}{k!l!}}{}_{\lambda }H_{k+l}^{\left(m\right)}(w,\rho ,\eta )=\sum _{p=0}^{\infty }{\displaystyle \frac{{(w-\xi )}^p{(t+\omega )}^p}{p!}}\sum _{k,l=0}^{\infty }{\displaystyle \frac{t^k{\omega }^l}{k!l!}}{}_{\lambda }H_{k+l}^{\left(m\right)}(\xi ,\rho ,\eta ).$$

This, when applying Equation (49) to the first expression of the right-hand side, yields

$$\sum _{k,l=0}^{\infty }{\displaystyle \frac{{\xi }^k{\omega }^l}{k!l!}}{}_{\lambda }H_{k+l}^{\left(m\right)}(w,\rho ,\eta )=\sum _{p,q=0}^{\infty }{\displaystyle \frac{{(w-\xi )}^{p+q}{t}^p{\omega }^q}{p!q!}}\sum _{k,l=0}^{\infty }{\displaystyle \frac{t^k{\omega }^l}{k!l!}}{}_{\lambda }H_{k+l}^{\left(m\right)}(\xi ,\rho ,\eta ).$$

(50)

Now, by replacing k by $k-p$, l by $l-q$ and using the following identity [21]:

$$\sum _{k=0}^{\infty }\sum _{p=0}^{\infty }A(p,k)=\sum _{k=0}^{\infty }\sum _{p=0}^{k}A(n,k-p),$$

(51)

in the right-hand side of Equation (50), we yield

$$\sum _{k,l=0}^{\infty }{\displaystyle \frac{t^k{\omega }^l}{k!l!}}{}_{\lambda }H_{k+l}^{\left(m\right)}(w,\rho ,\eta )=\sum _{k,l=0}^{\infty }\sum _{p,q=0}^{k,l}{\displaystyle \frac{{(w-\xi )}^{p+q}{t}^k{\omega }^l}{p!q!(k-p)!(l-q)!}}{}_{\lambda }H_{k+l-p-q}^{\left(m\right)}(\xi ,\rho ,\eta ).$$

(52)

Finally, comparing the equal powers of t and ω, we yield assertion (48). □

Remark 6. 

For $l=0$ and $w=\xi +z$, Equation (48) becomes Equation (44).

Remark 7. 

For $m=2$ in Theorem 6, we obtain the following given summation formula for Hermite–Lambda polynomials ${}_{\lambda }H_n(\xi ,\rho ,\eta )$:

$${}_{\lambda }H_{k+l}(w,\rho ,\eta )=\sum _{p,q=0}^{k,l}\left({\displaystyle \genfrac{}{}{0pt}{}{k}{p}}\right)\left({\displaystyle \genfrac{}{}{0pt}{}{l}{q}}\right){(w-\xi )}^{p+q}{}_{\lambda }H_{k+l-p-q}^{\left(m\right)}(\xi ,\rho ,\eta ).$$

(53)

Now, we obtain the following summation formula of GHLPs ${}_H\lambda _n^{\left(m\right)}(\xi ,\rho ,\eta )$:

Theorem 7. 

The given summation formula for the Gould–Hopper–Lambda polynomials ${}_H\lambda _n^{\left(m\right)}(\xi ,\rho ,\eta )$ is valid:

$${}_H\lambda _n^{\left(m\right)}(\xi +z,\eta +w,\eta )=\sum _{r=0}^n\left({\displaystyle \genfrac{}{}{0pt}{}{n}{r}}\right)H_{n-r}^{\left(m\right)}(\xi ,\rho ){}_H\lambda _r^{\left(m\right)}(z,w,\eta ).$$

(54)

Proof. 

Upon replacement of ξ with $\xi +z$ and ρ with $\rho +w$ in Equation (28), we yield

$$\sum _{n=0}^{\infty }{}_H\lambda _n^{\left(m\right)}(\xi +z,\eta +w,\eta ){\displaystyle \frac{t^n}{n!}}=exp\left((\xi +z)t+(\rho +w)t^m\right)cos\left(\sqrt{\eta t}\right),$$

which, upon using Equation (29), yields

$$\sum _{n=0}^{\infty }{}_H\lambda _n^{\left(m\right)}(\xi +z,\eta +w,\eta ){\displaystyle \frac{t^n}{n!}}=exp(\xi t+\rho t^m)exp\left(zt+w{t}^m\right)cos\left(\sqrt{\eta t}\right).$$

(55)

Making use of Equations (9) and (28) on the right-hand side of Equation (55), we obtain

$$\sum _{n=0}^{\infty }{}_H\lambda _n^{\left(m\right)}(\xi +z,\eta +w,\eta ){\displaystyle \frac{t^n}{n!}}=\sum _{n=0}^{\infty }\sum _{r=0}^{\infty }{H}_n^{\left(m\right)}(\xi ,\rho ){}_H\lambda _n^{\left(m\right)}(z,w,\eta ){\displaystyle \frac{t^n}{n!}}{\displaystyle \frac{t^r}{r!}},$$

which, upon making use of the series re-arrangement formula, becomes

$$\sum _{n=0}^{\infty }{}_H\lambda _n^{\left(m\right)}(\xi +z,\eta +w,\eta ){\displaystyle \frac{t^n}{n!}}=\sum _{n=0}^{\infty }\sum _{r=0}^n\left({\displaystyle \genfrac{}{}{0pt}{}{n}{r}}\right)H_{n-r}^{\left(m\right)}(\xi ,\rho ){}_H\lambda _r^{\left(m\right)}(z,w,\eta ){\displaystyle \frac{t^n}{n!}}.$$

(56)

Upon a comparison of equal powers of t from both sides of Equation (56), we obtain assertion (54). □

Remark 8. 

For $m=2$ in Theorem 7, we obtain the following summation formula for Hermite–Lambda polynomials ${}_{\lambda }H_n(\xi ,\rho ,\eta )$:

$${}_{\lambda }H_n(\xi +z,\eta +w,\eta )=\sum _{r=0}^n\left({\displaystyle \genfrac{}{}{0pt}{}{n}{r}}\right)H_{n-r}(\xi ,\rho ){}_{\lambda }H_r(z,w,\eta ).$$

(57)

4. Partial Differential Equations and Operational Identities

In this section, we have derived partial differential equations and operational identities for Gould–Hopper–Lambda polynomials.

We obtain the following recurrence relation of GHLPs ${}_H\lambda _n^{\left(m\right)}(\xi ,\rho ,\eta )$:

Theorem 8. 

The following recurrence relation is satisfied by Gould–Hopper–Lambda polynomials ${}_H\lambda _n^{\left(m\right)}(\xi ,\rho ,\eta )$:

$${}_H\lambda _{n+1}^{\left(m\right)}(\xi ,\rho ,\eta )=(\xi -\widehat{C}\eta ){}_H\lambda _n^{\left(m\right)}(\xi ,\rho ,\eta )+{\displaystyle \frac{n!}{(n-m+1)!}}my{}_H\lambda _{n-m+1}^{\left(m\right)}(\xi ,\rho ,\eta ).$$

(58)

Proof. 

Differentiating Equation (30) w.r.t t and then again using Equation (28) in the resultant equation, we have

$$\sum _{n=0}^{\infty }{}_H\lambda _n^{\left(m\right)}(\xi ,\rho ,\eta ){\displaystyle \frac{t^{n-1}}{(n-1)!}}=(\xi -\widehat{C}\eta )\sum _{n=0}^{\infty }{}_H\lambda _n^{\left(m\right)}(\xi ,\rho ,\eta ){\displaystyle \frac{t^n}{n!}}+my\sum _{n=0}^{\infty }{}_H\lambda _n^{\left(m\right)}(\xi ,\rho ,\eta ){\displaystyle \frac{t^{n+m-1}}{n!}},$$

which, upon comparing the equal powers of t from both sides of the above equation, gives assertion (8). □

Remark 9. 

For $m=2$ in Theorem 8, we obtain the following recurrence relation for Hermite–Lambda polynomials ${}_{\lambda }H_n(\xi ,\rho ,\eta )$:

$${}_{\lambda }H_{n+1}(\xi ,\rho ,\eta )=(\xi -\widehat{C}\eta ){}_{\lambda }H_n(\xi ,\rho ,\eta )+2ny{}_{\lambda }H_{n-1}(\xi ,\rho ,\eta ).$$

(59)

We obtain the following partial differential equation of GHLPs ${}_H\lambda _n^{\left(m\right)}(\xi ,\rho ,\eta )$:

Theorem 9. 

The Gould–Hopper–Lambda polynomials${}_H\lambda _n^{\left(m\right)}(\xi ,\rho )$satisfy the following partial differential equations:

$${\displaystyle \frac{\partial }{\partial \rho }}{}_H\lambda _n^{\left(m\right)}(\xi ,\rho ,\eta )={\displaystyle \frac{{\partial }^m}{\partial {\xi }^m}}{}_H\lambda _n^{\left(m\right)}(\xi ,\rho ,\eta ),$$

(60)

$${\displaystyle \frac{\partial }{\partial \eta }}{}_H\lambda _n^{\left(m\right)}(\xi ,\rho ,\eta )=-\widehat{C}{\displaystyle \frac{\partial }{\partial \xi }}{\lambda }_n^{\left(m\right)}(\xi ,\rho ,\eta ),$$

(61)

and

$${\displaystyle \frac{{\partial }^m}{\partial {\eta }^m}}{}_H\lambda _n^{\left(m\right)}(\xi ,\rho ,\eta )={(-1)}^m{\widehat{C}}^m{\displaystyle \frac{\partial }{\partial \rho }}{}_H\lambda _n^{\left(m\right)}(\xi ,\rho ,\eta ).$$

(62)

Proof. 

Differentiating Equation (28) with regard to ξ, we find

$${\displaystyle \frac{\partial }{\partial \xi }}\sum _{n=0}^{\infty }{}_H\lambda _n^{\left(m\right)}(\xi ,\rho ,\eta ){\displaystyle \frac{t^n}{n!}}=\sum _{n=0}^{\infty }{}_H\lambda _{n-m}^{\left(m\right)}(\xi ,\rho ,\eta ){\displaystyle \frac{t^{n+1}}{n!}},$$

which, upon differentiating Equation (28) with regard to ρ and then again using Equation (28) in the resultant equation, we yield

$${\displaystyle \frac{\partial }{\partial \rho }}{}_H\lambda _n^{\left(m\right)}(\xi ,\rho ,\eta )={\displaystyle \frac{n!}{(n-m)!}}{}_H\lambda _{n-m}^{\left(m\right)}(\xi ,\rho ,\eta ).$$

(63)

Again, differentiating Equation (9) m times with regard to ξ and then comparing the resultant equation with Equation (63), we obtain assertion (60).

Differentiating Equation (31) with regard to η and then again using Equation (31) in the resultant equation, we find

$${\displaystyle \frac{\partial }{\partial \eta }}\sum _{n=0}^{\infty }{}_H\lambda _n^{\left(m\right)}(\xi ,\rho ,\eta ){\displaystyle \frac{t^n}{n!}}=-\widehat{C}\sum _{n=0}^{\infty }{}_H\lambda _n^{\left(m\right)}(\xi ,\rho ,\eta ){\displaystyle \frac{t^{n+1}}{n!}},$$

(64)

by differentiating the above equation m times with regard to η, we obtain

$${\displaystyle \frac{{\partial }^m}{\partial {\eta }^m}}\sum _{n=0}^{\infty }{}_H\lambda _n^{\left(m\right)}(\xi ,\rho ,\eta ){\displaystyle \frac{t^n}{n!}}={(-1)}^m{\widehat{C}}^m\sum _{n=0}^{\infty }{}_H\lambda _n^{\left(m\right)}(\xi ,\rho ,\eta ){\displaystyle \frac{t^{n+m}}{n!}}.$$

(65)

Comparing equal powers of t from both the sides of Equations (64) and (65), we yield

$${\displaystyle \frac{\partial }{\partial \eta }}{}_H\lambda _n^{\left(m\right)}(\xi ,\rho ,\eta )=-n\widehat{C}{}_H\lambda _{n-1}^{\left(m\right)}(\xi ,\rho ,\eta ),$$

(66)

and

$${\displaystyle \frac{{\partial }^m}{\partial {\eta }^m}}{}_H\lambda _n^{\left(m\right)}(\xi ,\rho ,\eta )={(-1)}^m{\widehat{C}}^m{\displaystyle \frac{n!}{(n-m)!}}{}_H\lambda _{n-1}^{\left(m\right)}(\xi ,\rho ,\eta ),$$

(67)

respectively.

Now, comparing Equations (39) and (66), (63) and (67), we obtain assertions (61) and (62). □

Remark 10. 

For$m=2$in Theorem 9, we obtain the following partial differential equations for Hermite–Lambda polynomials${}_{\lambda }H_n(\xi ,\rho ,\eta )$:

$${\displaystyle \frac{\partial }{\partial \rho }}{}_{\lambda }H_n(\xi ,\rho ,\eta )={\displaystyle \frac{{\partial }^2}{\partial {\xi }^2}}{}_{\lambda }H_n(\xi ,\rho ,\eta ),$$

(68)

$${\displaystyle \frac{\partial }{\partial \eta }}{}_{\lambda }H_n(\xi ,\rho ,\eta )=-\widehat{C}{\displaystyle \frac{\partial }{\partial \xi }}{}_{\lambda }H_n(\xi ,\rho ,\eta ),$$

(69)

and

$${\displaystyle \frac{{\partial }^2}{\partial {\eta }^2}}{}_{\lambda }H_n(\xi ,\rho ,\eta )={\widehat{C}}^2{\displaystyle \frac{\partial }{\partial \rho }}{}_{\lambda }H_n(\xi ,\rho ,\eta ).$$

(70)

We obtain the following theorem for operational rules of GHLPs ${}_H\lambda _n^{\left(m\right)}(\xi ,\rho )$:

Theorem 10. 

The following operational identities are satisfied by Gould–Hopper–Lambda polynomials ${}_H\lambda _n^{\left(m\right)}(\xi ,\rho )$:

$${}_H\lambda _n^{\left(m\right)}(\xi ,\rho ,\eta )=e^{-{\widehat{C}}^{-1}z{D}_{\xi }}{H}_n^{\left(m\right)}(\xi ,\rho ),$$

(71)

and

$${}_H\lambda _n^{\left(m\right)}(\xi ,\rho ,\eta )=e^{{(-1)}^m{\widehat{C}}^{-m}y{D}_{\eta }^m}{\lambda }_n^{\left(m\right)}(\eta ,\xi ).$$

(72)

Proof. 

Solving Equations (60) and (62) with the initial conditions

$${}_H\lambda _n^{\left(m\right)}(\xi ,\eta ,0)=H_n^{\left(m\right)}(\xi ,\rho ),$$

and

$${}_H\lambda _n^{\left(m\right)}(\xi ,0,\eta )={\lambda }_n(\eta ,\xi ),$$

we obtain assertions (71) and (72). □

Remark 11. 

For $m=2$ in Theorem 10, we obtain the following operational rules for Hermite–Lambda polynomials ${}_{\lambda }H_n(\xi ,\rho ,\eta )$:

$${}_{\lambda }H_n(\xi ,\rho ,\eta )=e^{-{\widehat{C}}^{-1}z{D}_{\xi }}{H}_n(\xi ,\rho ),$$

(73)

and

$${}_H\lambda _n^{\left(m\right)}(\xi ,\rho ,\eta )=e^{{\widehat{C}}^{-2}y{D}_{\eta }^2}{\lambda }_n(\eta ,\xi ).$$

(74)

5. Conclusions and Future Work

The combination of umbral techniques with an umbral approach in the context of multi-variable special functions as well as polynomials offers a novel analytical framework for solving a high range of partial differential equations typically appeared in physical problems. The umbral symbolic method paves the way for introducing new hybrid families of special polynomials and functions with exploring their theoretical foundations. The primary motivation behind developing these techniques has been to address differential equations. These methods have gained significant popularity in the area of special function because of their versatility. These methods also inspire the formulation of new computer languages by using symbolic computation and manipulation [7,8,9].