Legendre-Gould Hopper-Based Sheffer Polynomials and Operational Methods

Various special polynomials have found diverse and vital applications in a number of fields such as mathematics, applied mathematics, mathematical physics, and engineering. According to the necessity of solving certain specific problems in diverse fields or pure mathematical interests, recently, a remarkably large number of new polynomials and numbers as well as a variety of generalizations (or extensions) and variants of some known polynomials have been established and investigated. In this regard, the Gould-Hopper polynomials incorporated with the Legendre polynomials were extended to be named as the Legendre-Gould Hopper polynomials by using operational methods. Also, certain Sheffer polynomials based on some known polynomials have been presented and investigated (see, e.g., [1,2,3,4]). In this paper, we aim to introduce the Legendre-Gould Hopper-based Sheffer polynomials by modifying the classical generating function (2) of the Sheffer polynomials. Since these newly introduced polynomials are found to be Sheffer type polynomials, we show that several properties and identities of the Sheffer polynomials are employed to give the corresponding results to these new polynomials. Moreover, other properties and formulas for these new polynomials such as quasi-monomials, other operational and certain integral representations are presented. We conclude that this method is pointed out to be easily applicable to other known polynomials, which are quasi-monomials with respect to some multiplicative and derivative operators, to yield certain Sheffer polynomials based on the known polynomials. In addition to this conclusion, the Legendre-Gould Hopper-based Sheffer polynomials generated by other generating functions with investigating their properties and formulas are poised as problems which are left to the interested researcher and the authors for future investigation.

For our purpose, some notations with modified ones and certain known facts are recalled and introduced. In what follows, let $\mathcal{P}$ denote the algebra of formal power series in the variable t over the field C of characteristic zero, say, $\mathbb{C}$ the field of complex numbers. If $f\left(t\right)$ and $g\left(t\right)$ in $\mathcal{P}$ satisfy then $f\left(t\right)$ and $g\left(t\right)$ are said to be a delta series and an invertible series, respectively. With each pair of a delta series $f\left(t\right)$ and an invertible series $g\left(t\right)$, there exists a unique sequence $s_n\left(x\right)$ of polynomials gratifying the orthogonality conditions (see [5], p. 17, Theorem 2.3.1) where ${\delta }_{n,k}$ is the Kronecker delta function defined by ${\delta }_{n,k}=1$ $(n=k)$ and ${\delta }_{n,k}=0$ $(n\ne k)$. The operator $〈·|·〉$ remains the same as in (see [5], Chapter 2). Here and elsewhere, let $\mathbb{N}$ be the set of positive integers and, also let ${\mathbb{N}}_0:=\mathbb{N}\cup \left\{0\right\}$. The sequence $s_n\left(x\right)$ satisfying (1) is called the Sheffer sequence for $\left(g\right(t),f(t\left)\right)$, or $s_n\left(x\right)$ is Sheffer for $\left(g\right(t),f(t\left)\right)$, which is usually denoted as $s_n\left(x\right)\sim (g\left(t\right),f\left(t\right))$. In particular, if $s_n\left(x\right)\sim (1,f\left(t\right))$, then $s_n\left(x\right)$ is called the associated sequence for $f\left(t\right)$, or $s_n\left(x\right)$ is associated with $f\left(t\right)$; if $s_n\left(x\right)\sim (g\left(t\right),t)$, then $s_n\left(x\right)$ is called the Appell sequence for $g\left(t\right)$, or $s_n\left(x\right)$ is Appell for $g\left(t\right)$ (see ([5], p. 17); see also [6,7]). Among diverse characterizations of Sheffer sequences, we recall the generating function (see, e.g., ([5], Theorem 2.3.4)): The sequence ${}_{{}_{[g,f]}}{s}_n\left(x\right)$ is Sheffer for $\left(g\right(t),f(t\left)\right)$ if and only if for all x in $\mathbb{C}$, where $\overline{f}\left(t\right)=f^{-1}\left(t\right)$ is the compositional inverse of $f\left(t\right)$. Here and elsewhere, the ordered-pair notation $[g,f]$ implies that the first element g is an invertible series and the second one f is a delta series.

A sequence $u_n\left(x\right)$ (deg $u_n\left(x\right)=n$) of polynomials is called Sheffer A-type zero if it has a generating function of the form (see, e.g., ([8], p. 222, Theorem 72); see also ([5], p. 19)). where $A\left(t\right)$ and $H\left(t\right)$ are an invertible series and a delta series, respectively. Thus, $s_n\left(x\right)$ is a Sheffer sequence if and only if $s_n\left(x\right)/n!$ is a sequence of Sheffer A-type zero. It is noted that sequences of Sheffer A-type zero were suggested to be named as poweroids by Steffensen ([9], p. 335), from which the concept of monomiality arose (see also ([5], p. 19)). The quasi-monomial treatment of special polynomials has been proved to be a powerful tool for the investigation of the properties of a wide class of polynomials such as Sheffer polynomials (see, e.g., [9,10,11,12]). A sequence of polynomials $p_n\left(x\right)$ $\left(n\in {\mathbb{N}}_0,x\in \mathbb{C}\right)$ of degree n is said to be quasi-monomials with respect to operators $\widehat{M}$ and $\widehat{P}$ (called, respectively, multiplicative and derivative operators) acting on polynomials if they do exist and satisfy It is easily found from (4) that and The algebra of $\widehat{P}$, $\widehat{M}$, the identity operator $\widehat{1}$ and the zero operator $\widehat{0}$ are seen to satisfy the commutation relations The algebra (7) is called the Heisenberg-Weyl algebra. The polynomials $p_n\left(x\right)$ are derived via the action of ${\widehat{M}}^n$ on $p_0\left(x\right)$: For $p_0\left(x\right)=1$, $p_n\left(x\right)={\widehat{M}}^n\left\{1\right\}$ and the exponential generating function of $p_n\left(x\right)$ is The simplest ones of (4) are the multiplication $\widehat{M}=X$ and derivative $\widehat{P}=D=\frac{d}{dx}$ operators acting on the space of polynomials. They act on monomials as follows: which lead to $[D,X]=1$ and, in view of linearity, are utilized to act on polynomials and formal power series.

For more details and applications of operational methods and quasi-monomials, one may be referred, for example, to [1,2,3,5,9,10,11,12,13,14,15,16,17,18,19,20,21].

We recall the Gould-Hopper polynomials (see ([22], Equation (6.2))) (also called sometimes as higher-order Hermite or Kampé de Fériet polynomials) $H_n^{\left(s\right)}(x,y)$ defined by A generating function of $H_n^{\left(s\right)}(x,y)$ is given by (see, e.g., ([22], Equation (6.3))) They appear as the solution of the generalized heat equation (see, e.g., [13]; see also ([23], p. 96)) Under the operational formalism (or a formal solution of (13) when it is considered to be a first-order differential equation with respect to the variable y with the initial condition), they are defined as These polynomials are quasi-monomial under the action of the operators (see [13])

We recall the Legendre polynomials $S_n(x,y)$ and $\frac{R_n(x,y)}{n!}$ which are defined by the generating functions (see, e.g., [24]): and respectively, where $C_0\left(x\right)$ denotes the 0th-order Bessel-Tricomi function. The nth order Bessel-Tricomi function $C_n\left(x\right)$ is defined by the following series (see, e.g., ([11], p. 150)): where $J_n\left(x\right)$ is the ordinary cylindrical Bessel function of the first kind (see, e.g., [23]). The operational definition of the 0th-order Bessel-Tricomi function $C_0\left(x\right)$ is given by (see, e.g., ([25], p. 86)) where $D_x^{-1}$ denotes the inverse of the derivative operator $D_x:=\frac{\partial }{\partial x}$ and is defined by means of the Riemann-Liouville fractional integral (see, e.g., ([26], p. 69)) It is easy to see that

Yasmin [25] introduced and studied the Legendre-Gould Hopper polynomials (LeGHP) ${}_S{H}_n^{\left(r\right)}(x,y,z)$ and $\frac{{}_R{H}_n^{\left(r\right)}(x,y,z)}{n!}$, which are defined, respectively, by the following generating functions: and The polynomials ${}_S{H}_n^{\left(r\right)}(x,y,z)$ and $\frac{{}_R{H}_n^{\left(r\right)}(x,y,z)}{n!}$ are quasi-monomials under the action of the multiplicative and derivative operators (see ([25], Equations (2.18a,b) and (2.19a,b))) and respectively.

In this section, Legendre-Gould Hopper-based Sheffer polynomials are introduced, and their quasi-monomial properties and differential equations are established.

The next two theorems show that the two polynomials introduced in Definitions 1 and 2 are quasi monomials for some derived multiplicative and derivative operators.

We begin with the following remark which depicts some simple properties about invertible and delta series.

The following theorems contain some properties which are easily derivable from those quasi monomials of those series given in Definitions 1 and 2.

For further assertions in the next sections, here is a suitable position to give some observations in the following remark.

In this section and in the sequel, unless otherwise stated, we consider the sequence ${}_{{}_{S}LeG{H}^{\left(r\right)}}{s}_n{(x,y,z)}_{[g,f]}$ as a function of the variable y and its associated sequence is Then, in view of Remark 4, the assertions of the theorems in this section can be verified by the corresponding theorems in ([5], pp. 17–24). In this regard, proofs are omitted.

The Equation (53) follows from the modified Equation (2.3.4) in ([5], p. 19) which should be corrected with replacing $f{\left(t\right)}^k$ by ${\left(f^{-1}\left(t\right)\right)}^k$.

For the sequence ${}_{{}_{S}LeG{H}^{\left(r\right)}}{s}_n{(x,y,z)}_{[g,f]}$ as a function of the variable y and its associated sequence is Then, in view of Remark 4, the assertions of the theorems in this section can be validated by the corresponding theorems in ([5], pp. 25–26). In this regard, proofs are omitted.