Volume 44, Issue 1

Many authors investigated the characteristics of the Bell, Euler, Bernoulli, and Genocchi polynomials because of their numerous uses in statistics, number theory, and other branches of science. A generating function for mixed-type Apostol–Euler polynomials of order η related with Bell polynomials is presented in this study. We also construct essential identities of Apostol–Euler polynomials of order η connected with Bell polynomials, such as the correlation formula, the implicit summation formula, the derivative formula, some association with Stirling numbers, and specific cases. Also we establish certain symmetric identities and their known consequences. In addition, we investigate several interesting relationships related to umbral calculus.

In this work, we propose some unified integral formulas for the Mittag-Leffler confluent hypergeometric function (MLCHF), and our findings are assessed in terms of generalized special functions. Additionally, certain unique cases of confluent hypergeometric function have been corollarily presented.

In this article, we get certain integral representations of the multi-index Wright generalized Bessel function by making use of the extended beta function. This function is presented as a part of the generalized Bessel–Maitland function obtained by taking the extended fractional derivative of the generalized Bessel–Maitland function developed by Özarsalan and Özergin [M. Ali Özarslan and E. Özergin, Some generating relations for extended hypergeometric functions via generalized fractional derivative operator, Math. Comput. Model. 52 2010, 9–10, 1825–1833]. In addition, we demonstrate the exciting connections of the multi-index Wright generalized Bessel function with Laguerre polynomials and Whittaker function. Further, we use the generalized Wright hypergeometric function to calculate the Mellin transform and the inverse of the Mellin transform.

In this paper we prove a new variant of q -Hermite–Hadamard–Mercer-type inequality for the functions that satisfy the Jensen–Mercer inequality (JMI). Moreover, we establish some new midpoint- and trapezoidal-type inequalities for differentiable functions using the JMI. The newly developed inequalities are also shown to be extensions of preexisting inequalities in the literature.

The introduction of two-parameter ( p , q ) {(p,q)} -calculus and Lie algebras in 1991 has spurred a wave of recent research into ( p , q ) {(p,q)} -special polynomials, including ( p , q ) {(p,q)} -Bernoulli, ( p , q ) {(p,q)} -Euler, ( p , q ) {(p,q)} -Genocchi and ( p , q ) {(p,q)} -Frobenius–Euler polynomials. These investigations have been carried out by numerous researchers in order to uncover a wide range of identities associated with these polynomials and applications. In this article, we aim to introduce ( p , q ) {(p,q)} -sine and ( p , q ) {(p,q)} -cosine based λ-array type polynomials and derive numerous properties of these polynomials such as ( p , q ) {(p,q)} -integral representations, ( p , q ) {(p,q)} -partial derivative formulae and ( p , q ) {(p,q)} -addition formulae. It is worth noting that the utilization of the ( p , q ) {(p,q)} -polynomials introduced in this study, along with other ( p , q ) {(p,q)} -polynomials, can lead to the derivation of various identities that differ from the ones presented here.