Identities Arising from a Certain Family of Special Combinatorial Numbers and Polynomials

Abstract

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 polynomials, including the Apostol–Euler, Apostol–Bernoulli, Frobenius–Euler, Fubini, and Stirling numbers. We also define an interpolation function for the higher-order numbers and show that its values at negative integers recover these numbers up to an explicit normalization factor. Furthermore, we obtain a residue-class decomposition of this interpolation function and express it in terms of the structured generalized hypergeometric Hurwitz–Lerch-type specialization arising from the residue-class decomposition. Some special cases and structural properties of this zeta-type function, including its relation to the classical Lerch transcendent and a differential identity, are also investigated. As applications, we show that for λ<−1 the interpolation function governs the moments of a zero-truncated negative binomial distribution naturally attached to the higher-order numbers, and we derive the asymptotic behavior of these numbers from the dominant singularity of the generating function.