We consider the N-tuple Hurwitz-Lerch zeta function introduced by Srivastava and Choi, ζN(s,x,λ)=∑k1,…,kN≥0λk1+⋯+kN(x+k1+⋯+kN)s, where N is a positive integer and λ is a complex parameter. A central breakthrough in this work is the explicit reduction of the N-tuple function to a linear combination of single Lerch transcendents via a finite-difference identity, which naturally leads to sharp multiplication and inversion formulas. We further investigate the combinatorial structure of the underlying polynomial coefficients, establishing their symmetry, unimodality, log-concavity, and precise asymptotic behavior. As key applications, we derive new identities for higher-order Apostol-Euler and Apostol-Bernoulli polynomials. Our results unify and extend several known formulas.
The Lerch transcendent is a three-parameter special function containing the Riemann zeta, Hurwitz zeta, Dirichlet eta, and polylogarithm functions as distinguished specializations. This study develops a defect-based framework for comparing fractional differentiation with parameter transformations and authentic function...
The results obtained in this paper extend several identities and interpolation formulas previously established for the numbers y9,n(λ;a) to the higher-order case. We derive several explicit formulas, identities, and recurrence-type relations, and establish connections with well-known families of special numbers and pol...
Damla Gun, A. Bayad, Yılmaz Şimşek· Mathematics· 0 citations
We establish logarithmic upper bounds for nonnegative multiplicative functions evaluated at values of multivariable polynomials. The polynomial contribution is encoded by the permutation character on the geometric irreducible components of the corresponding hypersurface, while the arithmetic contribution is described b...
We generalize the one-dimensional infinite-order linear differential equation satisfied by the $p$-adic Hurwitz-type Euler zeta function (Abh. Math. Semin. Univ. Hambg. 91: 117--135, 2021) to $n$ variables. By encoding the differential operator as a convolution with a distribution kernel, we introduce partial zeta func...
We study how the two cusp expansions of the Rogers-Ramanujan continued fraction control its ordinary and logarithmic Mellin moments. Ramanujan's factorizations express the logarithmic transform as a combination of products of zeta and quadratic Dirichlet $L$-functions. Its adjacent integer and half-integer values give...
In this paper, we will study a trivariate extension of the Cauchy numbers of both the first kind (also called Gregory coefficients) and the second kind (also called N\"orlund numbers) via the Laurent expansion of the reciprocal of any positive integer power (which is called the order) of multiple polylogarithms. In the...
Ce Xu, Jian-Qiang Zhao· 0 citations
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