A New Class of Higher-order Hypergeometric Bernoulli Polynomials Associated with Hermite Polynomials

Waseem A. Khan abstract: In this paper, we introduce a new class of higher-order hypergeometric Hermite-Bernoulli numbers and polynomials. We shall provide several properties of higher-order hypergeometric Hermite-Bernoulli polynomials including summation formulae, sums of product identity, recurrence relations.


Introduction
The Bernoulli polynomials B n (x) are defined by the following generating function (see [ [19], p.5, (2.2)]). They have many applications in modern number theory, such as modular forms [11] and Iwasawa theory [9]. A recent book by Arakawa, Ibukiyama and Kaneko [1] give a nice introduction of Bernoulli numbers and polynomials including their connections with zeta functions.
For N, r ∈ N, the higher-order hypergeometric Bernoulli polynomials B (r) N,n (x) are defined by means of the generating function, (see [2], [7], [10]) N,n (0) are called the higher order hypergeometric Bernoulli numbers, (see [10], [13]). Again, on taking r = 1 in (1.6), B It is easily seen that where H n (x) and He n (x) are called the ordinary Hermite polynomials. Also The generating function for Hermite polynomial H n (x,y) ( [16]- [18]) are given by H n (x, y) t n n! . (1.8) A New Class of Higher-order Hypergeometric Bernoulli...

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The object of this paper is to present a systematic account of these families in a unified and generalized form. We develop some elementary properties and derive the implicit summation formulae for the higher-order hypergeometric Hermite-Bernoulli polynomials by using different analytical means on their respective generating functions. The approach given in recent papers of Pathan and Khan ( [16]- [18]) has indeed allowed the derivation of implicit summation formulae in the two-variable higher-order hypergeometric Hermite-Bernoulli polynomials. In addition to this, some relevant connections between Hermite and higher-order hypergeometric Bernoulli polynomials and recurrence relations are given.

Multiple hypergeometric Hermite-Bernoulli numbers and polynomials
For every positive integer N and r, the higher-order hypergeometric Hermite-Bernoulli numbers and polynomials H B (r) N,n (x, y) are defined by means of the following generating function defined in a suitable neighborhood of t = 0: N,n (0, 0) are called the higher-order hypergeometric Bernoulli numbers, (see [10,13]). When r = 1, we obtain the hypergeometric Hermite-Bernoulli polynomials H B N,n (x, y) = H B N,n (0, 0) is the hypergeometric Bernoulli numbers, (see [8,15]). If we put N = 1, the result reduces to the known result of Pathan and Khan, (see [16]).
Remark 2.1. On setting y = 0, (2.1) reduces to the known result of Aoki et al. [2] as follows: In particular in terms of higher-order hypergeometric Bernoulli numbers B  Using e it = cost + i sin t and N = 1, the result reduces to and where Ω = (cos t − 1) 2 + (sin t) 2 , together with the definition (2.1) and the result (2.5), we get (see Pathan and Khan [16]): On setting r = 1, x = y = 0 in the above results , we get the following well known classical results involving Bernoulli numbers, (see [16]): Comparing the coefficients of t n on both sides, we get (2.7). ✷ and r = 1 in (2.8), the result reduces to (see [7]): Theorem 2.5. The following relationship holds true: Proof. Using equations (2.1), (1.5) and (1.8), we have Comparing the coefficients of t n n! on both sides, we arrive at the obtained result (2.11). ✷ Theorem 2.6. The following relationship holds true: Proof. From (2.1), we have Therefore, by integrating (2.13) with weight (1 − x) N −1 and using the result ( [20], p.26(48)), we obtain which follows from (2.12). This completes the proof. ✷ Proof. Using generating function for Hermite-Euler polynomials as follows t n n! , (see [18]).
Substituting this value of e xt+yt 2 in (2.1) gives Comparing the coefficients of t n n! on both sides, we required at the result (2.14). ✷ Theorem 2.8. For n ≥ 0, p, q ∈ R, the following formula for higher-order hypergeometric Hermite-Bernoulli polynomials H B (r) N,n (px, qy) holds true: Proof. Rewrite the generating function (2.1), we have Replacing k by k − 2j in above equation, we have Again replacing n by n − k in the above equation, we have Finally, equating the coefficients of t n on both sides, we acquire the result (2.15). ✷ Theorem 2.9. For n ≥ 0, p, q ∈ R and x, y ∈ C, we have H B

Summation formulae for higher-order hypergeometric Hermite-Bernoulli polynomials
In this section, we derive the summation formula, the sum of the product of identity and recurrence relations. First, we prove the following results involving higher-order hypergeometric Hermite-Bernoulli polynomials H B Proof. We replace t by t + u and rewrite the generating function (2.1) as Replacing x by z in the above equation and equating the resulting equation to the above equation, we get On expanding exponential function (3.3) gives which on using formula ( [20], p.52 (2)) (3.5) in the left hand side becomes ∞ n,p=0 Now replacing k by k − n, l by l − p and using the lemma ( [20], p.100 (1)) in the left hand side of (3.6), we get Finally on equating the coefficients of the like powers of t and u in the above equation, we get the required result. ✷ Proof. By exploiting the generating function (2.1) and using (1.8), we can write equation (2.1) as Replacing n by n − m in above equation and using lemma ( [20], p.101(1)), we get  Proof. We replace x by x + z and y by y + u in (2.1), use (1.2) and rewrite the generating function as Comparing the coefficients of t on both sides, we get the result (3.14). ✷ Proof. We replace x by y and y by x in equation (2.1) to get Replacing n by n − m in above equation and comparing the coefficients of t, we obtain (3.16). On replacing z by z − α − x and u by u − β + y in (3.16), we get (3.17). ✷ and Proof. Consider the definition of (2.1), we have Using the Cauchy product and comparing the coefficients of t, we obtain (3.18). Another way of defining the right hand side of equation ( Using the Cauchy product and comparing the coefficients of t, we get (3.19). ✷