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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.

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.

Many authors defined and extended the beta function in various forms because the beta function has wide uses in different fields of science and applied science. In this article, we define a new more generalized form of the extended beta matrix function via the Wiman matrix function and describe their significant properties and special cases. Furthermore, we define an extension of the Gauss hypergeometric and confluent hypergeometric matrix functions by adopting a novel type of beta matrix function. We also derive their Laplace transform, derivative formula and transformation formulae.

In this present paper, we define a generating function of mix type polynomials known as Bell based Apostol-Bernoulli polynomials of order γ. Furthermore, we define some important properties and identities such as implicit summation formula, correlation formula and their partial derivatives formulas. Also, we define some relation with stirling numbers.

A variety of functions, their extensions, and variants have been extensively investigated, mainly due to their potential applications in diverse research areas. In this paper, we aim to introduce a new extension of Whittaker function in terms of multi-index confluent hypergeometric function of first kind. We discuss multifarious properties of newly defined multi-index Whittaker function such as integral representation, integral transform (i.e., Mellin transform and Hankel transform), and derivative formula. The results presented here, being very general, are pointed out to reduce to yield some known or new formulas and identities for relatively functions.

In the present article, we study Bell based Euler polynomials of order αand investigate some correlation formula, summation formula and derivative formula. Also, we introduce some relations of Stirling numbers of the second kind. Moreover, we derive several important formulae of Bell based Euler polynomials by using umbral calculus.