On the new bicomplex generalization of Hurwitz-Lerch zeta function with properties and applications

In the recent years, various authors introduced different generalizations of the Hurwitz-Lerch zeta function and discussed its various properties. The main aim of our study is to introduce a new bicomplex generalization of the Hurwitz-Lerch zeta function using the new generalized form of the beta function that involves the Appell series and Lauricella functions. The new bicomplex generalization of the Hurwitz-Lerch zeta function reduces to some already known functions like the Hurwitz-Lerch zeta function, Hurwitz zeta function, Riemann zeta function and polylogarithmic function. Its different properties such as recurrence relation, summation formula, differentiation formula, generating relations and integral representations are investigated. All results induced are general in nature and reducible to already known results. As an application of the new bicomplex generalization of the Hurwitz-Lerch zeta function, we have developed a new generalized form of fractional kinetic equation and obtained its solution using the natural transform.