Search

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]