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The introduction of two-parameter (p , q) -calculus and Lie algebras in 1991 has spurred a wave of recent research into (p , q) -special polynomials, including (p , q) -Bernoulli, (p , q) -Euler, (p , q) -Genocchi and (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) -sine and (p , q) -cosine based λ-array type polynomials and derive numerous properties of these polynomials such as (p , q) -integral representations, (p , q) -partial derivative formulae and (p , q) -addition formulae. It is worth noting that the utilization of the (p , q) -polynomials introduced in this study, along with other (p , q) -polynomials, can lead to the derivation of various identities that differ from the ones presented here. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

In this research article, we provide an extended Hurwitz–Lerch zeta function using the extended beta function. We have also investigated the new generalized Hurwitz–Lerch zeta function's exciting characteristics, such as integral representations, differential equations, the Mellin transform and special generating relations. In order to support the main result, we have additionally considered several notable specific cases. Furthermore, we have discussed the solution of the fractional kinetic equation using the new extended Hurwitz–Lerch zeta function. [ABSTRACT FROM AUTHOR]