Mixed-Type Hypergeometric Bernoulli–Gegenbauer Polynomials

: In this paper, we consider a novel family of the mixed-type hypergeometric Bernoulli– Gegenbauer polynomials. This family represents a fascinating fusion between two distinct categories of special functions: hypergeometric Bernoulli polynomials and Gegenbauer polynomials. We focus our attention on some algebraic and differential properties of this class of polynomials, including its explicit expressions, derivative formulas, matrix representations, matrix-inversion formulas, and other relations connecting it with the hypergeometric Bernoulli polynomials. Furthermore, we show that unlike the hypergeometric Bernoulli polynomials and Gegenbauer polynomials, the mixed-type hypergeometric Bernoulli–Gegenbauer polynomials do not fulﬁll either Hanh or Appell conditions.

The polynomials V A significant amount of research has been conducted on various generalizations and analogs of the Bernoulli polynomials and the Bernoulli numbers.For a comprehensive treatment of the diverse aspects, including summation formulas and applications, interested readers can refer to recent works [1,2].Inspired by recent articles [3][4][5][6][7] where authors explore analytic and numerical aspects of hypergeometric Bernoulli polynomials, hypergeometric Euler polynomials, generalized mixed-type Bernoulli-Gegenbauer polynomials, and Lagrange-based hypergeometric Bernoulli polynomials, this article focuses on the algebraic and differential properties of the polynomials . These properties include their explicit expressions, derivative formulas, matrix representations, matrix-inversion formulas, and other relationships connecting them with hypergeometric Bernoulli polynomials.
The paper is organized as follows.Section 2 provides relevant information about hypergeometric Bernoulli polynomials and Gegenbauer polynomials.Section 3 is dedicated to the study of the main algebraic and analytic properties of the HBG polynomials (1) and (2), which are summarized in Theorems 1-4, and Proposition 6.

Background and Previous Results
Throughout this paper, let N, N 0 , Z, R, and C denote, respectively, the sets of natural numbers, non-negative integers, integers, real numbers, and complex numbers.As usual, we always use the principal branch for complex powers, in particular, 1 α = 1 for α ∈ C. Furthermore, the convention 0 0 = 1 is adopted.
For λ ∈ C and k ∈ Z, we use the notations λ (k) and (λ) k for the rising and falling factorials, respectively, i.e., and From now on, we denote by P n the linear space of polynomials with real coefficients and a degree less than or equal to n.Moreover, to present some of our results, we require the use of the generalized multinomial theorem (cf.[8,9] and the references therein).

Hypergeometric Bernoulli Polynomials
For a fixed m ∈ N, the hypergeometric Bernoulli polynomials are defined by means of the following generating function [5,[10][11][12][13][14]: and the hypergeometric Bernoulli numbers are defined by (0) for all n ≥ 0. The hypergeometric Bernoulli polynomials also are called generalized Bernoulli polynomials of level m [5,6].It is clear that if m = 1 in (3), then we obtain the definition of the classical Bernoulli polynomials B n (x) and classical Bernoulli numbers, respectively, i.e., n , respectively, for all n ≥ 0. The first four hypergeometric Bernoulli polynomials are as follows: The following results summarize some properties of the hypergeometric Bernoulli polynomials (cf.[5,6,11,12,15]).
(e) Integral formulas. x (f) ( [12], Theorem 3.1) Differential equation.For every n ≥ 1, the polynomial B As a straightforward consequence of the inversion Formula (6), the following expected algebraic property is obtained.Proposition 2 ([5], Proposition 2).For a fixed m ∈ N and each n ≥ 0, the set B is a basis for P n , i.e., Let ζ(s) be the Riemann zeta function defined by The following result provides a formula for evaluating ζ(2r) in terms of the hypergeometric Bernoulli numbers.Proposition 3 ([6], Theorem 3.3).For a fixed m ∈ N and any r ∈ N, the following identity holds. where

Gegenbauer Polynomials
where the constant M α n is positive.It is clear that the normalization above does not allow α to be zero or a negative integer.Nevertheless, the following limits exist for every x ∈ [−1, 1] (see [16], (4.7.8)) where T n (x) is the nth Chebyshev polynomial of the first kind.In order to avoid confusing notation, we define the sequence { Ĉ(0) n (x)} n≥0 as follows: We denote the nth monic Gegenbauer orthogonal polynomial by where the constant k α n (cf.[16], Formula (4.7.31)) is given by Then for n ≥ 1, we have Gegenbauer polynomials are closely connected with axially symmetric potentials in n dimensions (cf.[4] and the references cited therein), and contain the Legendre and Chebyshev polynomials as special cases.Furthermore, they inherit practically all the formulas known in the classical theory of Legendre polynomials.
(f) For every n ∈ N (see [18], Proposition 2.1) As is well known, the monic Gegenbauer orthogonal polynomials admit other different definitions [16,[19][20][21].In order to deal with the definitions (1) and (2) of the HBG polynomials, we also are interested in the definition of the monic Gegenbauer orthogonal polynomials by means of the following generating functions: and 2π − xz Remark 1.Note that (10) and (11) are suitable modifications of the generating functions for the Gegenbauer polynomials Ĉ(α) n (x):

The Polynomials V
[m−1,α] n (x) and Their Properties Now, we can proceed to investigate some relevant properties of the HBG polynomials.
be the sequence of HBG polynomials of order α.Then the following explicit formulas hold.
Proof.On account of the generating functions ( 1) and (10), it suffices to make a suitable use of Cauchy product of series in order to deduce the expression (12).
Similarly, taking into account the generating functions ( 2) and (11), we can use an analogous reasoning to the previous one to obtain expression (13).
Thus, the suitable use of ( 8) and ( 12) allow us to check that for α ∈ (−1/2, ∞) \ {0}, the first five HBG polynomials are: In contrast to the hypergeometric Bernoulli polynomials and Gegenbauer polynomials, the HBG polynomials neither satisfy a Hanh condition nor an Appell condition.More precisely, we have the following result.
Furthermore, it is possible to establish an integral formula connecting the HBG polynomials with the monic Gegenbauer polynomials.This integral formula allows us to deduce a concise expression for the Fourier coefficients of the HBG polynomials in terms of the basis of monic Gegenbauer polynomials.
be the sequence of HBG polynomials of order α.Then, the following formula holds.
Regarding the zero distribution of these polynomials, the numerical evidence indicates that this distribution does not align with the behavior of either Bernoulli hypergeometric polynomials or Gegenbauer polynomials.For instance, in Figure 1, the plots for the zeros of V  As expected, the symmetry property of Gegenbauer polynomials is not inherited by the HBG polynomials.For instance, Figure 2 displays the induced mesh of V   For any α ∈ (−1/2, ∞), it is possible to deduce interesting relations connecting the HBG polynomials V
Proof.From the identities (2) and (3), we have Multiplying, respectively, the left-hand side of the above expression by (2π − xz) and the right-hand side by 1 − xz π + z 2 4π 2 , we obtain the following equivalent expression: Therefore, by comparing the coefficients on both sides of (21), we obtain the identities (20).
and from (7): It is clear that the matrix C (α) (x) is a lower triangular matrix for each x ∈ R, so that det C (α) (x) = 1.Therefore, C (α) (x) is a nonsingular matrix for each x ∈ R and α ∈ (−1/2, ∞). where Proof.For each r = 0, 1, . . ., n, consider the matrix form (24) of where That is, the entries of the matrix B(x) are the first four classical Bernoulli polynomials.
We can now proceed as outlined in [5].From the summation Formula (4) it follows and the differential equation presented in part (f) of Proposition 1 (cf.[12], Theorem 3.1) suggest that the HBG polynomials satisfy a differential equation of order n.These two properties, along with their implications and potential applications, will be the focus of our future work.Data Availability Statement: Data sharing is not applicable to this article.
fusion between two classes of special functions: hypergeometric Bernoulli polynomials and Gegenbauer polynomials.

Theorem 2 .
For a fixed m ∈ N, the HBG polynomials V [m−1,0] n (x) are related with the hypergeometric Bernoulli polynomials B [m−1] n (x) by means of the following identities. 2πB

Author Contributions:Funding:
Conceptualization, D.P. and Y.Q.; methodology, D.P. and Y.Q.; formal analysis, D.P., Y.Q. and S.A.W.; investigation, D.P., Y.Q. and S.A.W.; writing-original draft preparation, Y.Q.; writing-review and editing, D.P., Y.Q. and S.A.W.; supervision, Y.Q.; project administration, Y.Q. and S.A.W.; funding acquisition, Y.Q.All authors have read and agreed to the published version of the manuscript.The research of Y. Quintana has been partially supported by the grant CEX2019-000904-S funded by MCIN/AEI/10.13039/501100011033,and by the Madrid Government (Comunidad de Madrid-Spain) under the Multiannual Agreement with UC3M in the line of Excellence of University Professors (EPUC3M23), in the context of the Fifth Regional Programme of Research and Technological Innovation (PRICIT).