Legendre-Gould Hopper-Based Sheffer Polynomials and Operational Methods

: A remarkably large of number of polynomials have been presented and studied. Among several important polynomials, Legendre polynomials, Gould-Hopper polynomials, and Sheffer polynomials have been intensively investigated. In this paper, we aim to incorporate the above-referred three polynomials to introduce the Legendre-Gould Hopper-based Sheffer polynomials by modifying the classical generating function of the Sheffer polynomials. In addition, we investigate diverse properties and formulas for these newly introduced polynomials.


Introduction and Preliminaries
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 P denote the algebra of formal power series in the variable t over the field C of characteristic zero, say, C the field of complex numbers. If f (t) and g(t) in P satisfy then f (t) and g(t) are said to be a delta series and an invertible series, respectively. With each pair of a delta series f (t) and an invertible series g(t), there exists a unique sequence s n (x) of polynomials gratifying the orthogonality conditions (see [5], p. 17, Theorem 2.3.1) where δ n,k is the Kronecker delta function defined by δ n,k = 1 (n = k) and δ n,k = 0 (n = k).
The operator · | · remains the same as in (see [5], Chapter 2). Here and elsewhere, let N be the set of positive integers and, also let N 0 := N ∪ {0}. The sequence s n (x) satisfying (1) is called the Sheffer sequence for (g(t), f (t)), or s n (x) is Sheffer for (g(t), f (t)), which is usually denoted as , t), then s n (x) is called the Appell sequence for g(t), or s n (x) is Appell for g(t) (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 for all x in C, wheref (t) = f −1 (t) is the compositional inverse of f (t). 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 (x) (deg u n (x) = 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(t) and H(t) are an invertible series and a delta series, respectively. Thus, s n (x) is a Sheffer sequence if and only if s n (x)/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 (x) (n ∈ N 0 , x ∈ C) of degree n is said to be quasi-monomials with respect to operatorsM andP (called, respectively, multiplicative and derivative operators) acting on polynomials if they do exist and satisfŷ It is easily found from (4) thatMP {p n (x)} = n p n (x) andPM {p n (x)} = (n + 1) p n (x).
The algebra ofP,M, the identity operator1 and the zero operator0 are seen to satisfy the commutation relations The algebra (7) is called the Heisenberg-Weyl algebra. The polynomials p n (x) are derived via the action ofM n on p 0 (x): For p 0 (x) = 1, p n (x) =M n {1} and the exponential generating function of p n (x) is The simplest ones of (4) are the multiplicationM = X and derivativeP = D = d dx operators acting on the space of polynomials. They act on monomials as follows: Xx n = x n+1 and Dx n = nx n−1 , 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 A generating function of H (s) n (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 R n (x,y) n! which are defined by the generating functions (see, e.g., [24]): and respectively, where C 0 (x) denotes the 0th-order Bessel-Tricomi function. The nth order Bessel-Tricomi function C n (x) is defined by the following series (see, e.g., ( [11], p. 150)): where J n (x) 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 (x) is given by (see, e.g., ( [25], p. 86)) where D −1 x denotes the inverse of the derivative operator D x := ∂ ∂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) n (x, y, z) and R H (r) n (x,y,z) n! , which are defined, respectively, by the following generating functions: and The polynomials S H (r) n (x, y, z) and R H (r) n (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 and respectively.

Legendre-Gould Hopper Based Sheffer polynomials
In this section, Legendre-Gould Hopper-based Sheffer polynomials are introduced, and their quasi-monomial properties and differential equations are established.
for the functions g and f are defined by the following generating function where f (t) and g(t) are a delta series and an invertible series, respectively.

Remark 1.
The Sheffer-type polynomials in Definition 1 are well-defined in the following sense: We find from (24) that Furthermore, in view of (2), we can write Equating the last series in (27) and (28) and comparing the coefficients of t n on both sides of the resulting identity gives Also, in view of (22), we have Definition 2. The Legendre-Gould Hopper-based Sheffer polynomials R LeGH (r) s n (x,y,z) [g, f ] n! for the functions g and f are defined by the following generating function where f (t) and g(t) are a delta series and an invertible series, respectively.

Remark 2.
As in Remark 1, the Sheffer-type polynomials in Definition 2 are also well-defined in the following sense: Indeed, it follows from (25) that Moreover, in terms of (2), we get Identifying the last series in (31) and (33) and equating the coefficients of t n on both sides of the resultant identity yields Also, in view of (23), we find 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. and respectively. Here and elsewhere ∂ y := ∂ ∂y .
Proof. We find Since f −1 denotes the compositional inverse of the function f and f (t) has a series expansion in t, we have Differentiating both sides of the second equality in (27) with respect to t affords Since f (t) is a delta series in t, f (t) becomes an invertible series in t. Therefore 1 f ( f −1 (t)) can be a series expansion of f −1 (t). Since g(t) is an invertible series in t, 1 g( f −1 (t)) can be a series expansion of f −1 (t). It is found from (26) that F [g, f ] (x, y, z : t) is a product of 1 g( f −1 (t)) and another function which can be expanded in powers of f −1 (t). So we observe that F [g, f ] (x, y, z : t) can be a series expansion in powers of f −1 (t).
Using (39), we get Employing the series definition of (26) in Equation (43) provides both sides of which, upon equating the coefficients of t n , gives Therefore Equation (44) fulfills the second identity of (4) for quasi monomials. and respectively. Here and elsewhere ∂ x := ∂ ∂x .
Proof. The proof would run parallel with that of Theorem 1. The involved details are omitted.

Results Derivable from the Previous Section
We begin with the following remark which depicts some simple properties about invertible and delta series.
Remark 3. Let f (t) and g(t) be delta and invertible series, respectively, whose power series forms are and where {a k } k≥2 and {b k } k∈N are sequences of constants in C. Then the following properties are satisfied: (i) f (0) = 0, f (0) = 1 and g(0) = 1; (ii) The compositional inverse f −1 (t) of f (t) exists (for any delta series) and is of the same form as in (47); (iii) The function 1 g(t) is defined (for any invertible series) and is of the same form as in (48).
The following theorems contain some properties which are easily derivable from those quasi monomials of those series given in Definitions 1 and 2.
Theorem 3. Let f (t) and g(t) be delta and invertible series, respectively, whose power series forms are the same as in (47) and (48). Also let n ∈ N 0 . Then we find (c) A differential equation is Proof. The assertions here follow immediately from the results in Theorem 1 by using (5), (8), and (9).
Theorem 4. Let f (t) and g(t) be delta and invertible series, respectively, whose power series forms are the same as in (47) and (48). Also let n ∈ N 0 . Then we find (a) R LeGH (r) s 0 (x, y, z) [g, f ] = 1 and so Proof. The assertions here follow immediately from the results in Theorem 2 by using (5), (8), and (9).
For further assertions in the next sections, here is a suitable position to give some observations in the following remark.

Remark 4.
(a) The operator ·| · in (1) can be expressed as follows: More generally, for f (t) ∈ P and p(x) being a series in the variable x, we find In view of ( [5] p. 20, Theorem 2.3.7), the identity (44) implies that the sequence S LeGH (r) s n−1 (x, y, z) [g, f ] is Sheffer for (g(t), f (t)), when the sequence is looked upon as a function of the variable y.

Sheffer Sequences
In this section and in the sequel, unless otherwise stated, we consider the sequence S LeGH (r) 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][18][19][20][21][22][23][24]. In this regard, proofs are omitted. Theorem 5. The following identity holds. Theorem 6. For any h(t) ∈ P, Theorem 7. For any polynomial p(y), Theorem 8. The following expansion holds. (54) for all u ∈ C.
Theorem 11. For any h(t) and l(t) in P, (56)

Associated
For the sequence S LeGH (r) 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.

Theorem 12. The generating function is
Theorem 13. For any h(t) ∈ P, Theorem 14. For any polynomial p(y), Theorem 15. The conjugate representation is Theorem 16. For any h(t) ∈ P, Theorem 17. The binomial identity is for all u in C.

Appell Sequences
For the sequence S LeGH (r) s n (x, y, z) [g, f ] as a function of the variable y, its Appell sequence is Then, in view of Remark 4, the assertions of the theorems in this section can be justified by the corresponding theorems in ( [5], pp. [26][27][28]. In this respect, proofs are omitted.

Theorem 18. The generating function is
Theorem 19. For any h(t) ∈ P, Theorem 20. For any polynomial p(y), where p (k) (y) is the kth derivative of p(y).

Example 3. From
and t] t n (n!) 2 . (79) Using (22) and (23) in (78) and (79), respectively, gives and In addition, the sequence S LeGH (r) s n (x, y, z) [e t ,t] is found to belong to Appell sequence. Therefore the results in Section 6 may apply to this sequence to yield more involved identities. For example, S LeGH (r) s n (x, y + u, z) [e t ,t] = n ∑ k=0 n k S LeGH (r) s k (x, u, z) [e t ,t] y n−k . (82) Then it is easy to see that g is an entire function and an invertible series in P. Now use f (t) = t and g(t) = g(t) in (26) and set x = z = 0, and replace y by x. We get ∞ ∑ n=0 S LeGH (r) s n (0, x, 0) [g,t] t n n! = t e xt e t − 1 = ∞ ∑ n=0 B n (x) t n n! (|t| < 2π).
Or, equivalently, B n (x + y) = n ∑ k=0 n k B k (x) y n−k (n ∈ N 0 ) , which is a well known identity (see, e.g., ([27] p. 82, Equation (13))). and where A(t) is an invertible series and H(t) is a delta series, we are believed to be able to establish corresponding results as those in this paper. This posed problem is left to the interested researcher and the authors for future investigation.