Roberto Bagsarsa Corcino

This study introduces $(α,a)$-parameterized Hurwitz-Lerch type poly-Bernoulli and poly-Cauchy numbers and polynomials, extending classical sequences through the Hurwitz-Lerch zeta function. We derive generating functions, recurrences, and explicit formulas, revealing deeper structures and connections with zeta functions and polylogarithms. The results enrich existing theory and open new directions in analytic number theory and combinatorics.

This paper introduces a novel variation of Frobenius-Euler poly- nomials derived from Bell numbers and Apostol-type functions, in- corporating the polylogarithm concept. We explore their structure and properties through various analytical methods, with a particular emphasis on generating functions designed for higher-order Apostol- Frobenius-Type Poly-Euler polynomials based on Bell numbers. These functions facilitate the derivation of both explicit and implicit sum- mation formulas. Furthermore, we establish symmetric identities that unveil intricate polynomial relationships. This integration provides a new framework that deepens the understanding and expands the applicability of Poly-Euler polynomials. Our findings contribute to combinatorial and algebraic mathematics, fostering further research in related areas.

In this paper, we present and prove a novel expression for binomial sums involving generalized hyperharmonic numbers. Our approach utilizes Euler's transformation applied to the ordinary generating function of the generalized hyperharmonic numbers. To demonstrate the relevance of this new expression, we derive several identities that reveal connections between the characteristic equations and Binet forms of notable numerical sequences, including the Fibonacci, Lucas, Pell, Pell-Lucas, Jacobsthal, Jacobsthal-Lucas, Mersenne, and Mersenne-Lucas numbers. Furthermore, we establish the integer power representation of the generalized hyperharmonic sums. As an extension of our findings, we also introduce and prove an alternative expression using the polylogarithmic form of the generating function for the generalized hyperharmonic numbers.

This paper introduces a novel variation of poly-Genocchi polyno- mials by combining elements from the modified degenerate polyexpo- nential function, Hermite-Based Apostol-Genocchi polynomials, and Frobenius polynomials. These newly formulated polynomials, termed as the degenerate Hermite-Based Apostol-Frobenius-type poly-Genocchi polynomials with parameters a, b, and c, are explored to derive various identities and formulas. These include recurrence relations, explicit formulas, and specific differential identities. Furthermore, the pa- per establishes connections between these polynomials and degenerate Stirling numbers of the first and second kind, higher-order degenerate Bernoulli polynomials, and higher-order degenerate Frobenius-Euler polynomials.

This paper introduces and investigates a novel class of combinatorial constructs termed the $r$-Stirling Fibonacci numbers and polynomials of the first and second kind. By integrating the exponential generating functions of classical Fibonacci numbers with those of the signed $r$-Stirling numbers, we establish an enriched algebraic framework that advances the theory of special numbers and polynomials. The study yields new identities - including horizontal generating functions, explicit formulas, and convolution relations - that extend classical combinatorial results. In addition, we define the $r$-Stirling Chebyshev polynomials of both kinds by employing hyperbolic functions and exponential techniques, thereby forging a functional link between Fibonacci-type and Stirling-type sequences. These results are rigorously validated through series expansion and the Cauchy product method. The theoretical contributions of this work highlight the interplay between combinatorics, algebra, and analysis, with potential applications in number theory, orthogonal polynomials, and symbolic computation.

Background: This paper introduces a class of special polynomials called Type 2 degenerate poly-Frobenius-Euler polynomials, defined using the polyexponential function. Motivated by the expanding theory of degenerate versions of classical polynomials, the paper seeks to enrich the mathematical landscape by constructing generalized structures with deeper combinatorial and analytic properties. Methods: The study employs the method of generating functions combined with Cauchy's rule for the product of two series to derive explicit formulas and identities, enabling systematic manipulation of series expansions. From an analytic perspective, the authors utilized the comparison test and principles of uniform convergence to establish that certain integral representations correspond to holomorphic functions. Results: The researchers successfully derived explicit formulas and identities for the Type 2 degenerate poly-Frobenius-Euler polynomials. They established meaningful connections with the degenerate Stirling numbers of the first and second kinds. Furthermore, they introduced the Type 2 degenerate unipoly-poly-Frobenius-Euler polynomials, defined via the unipoly function, and thoroughly investigated their various properties, including behaviors under differentiation and integration. Conclusion: The study significantly advances the theory of degenerate polynomials by constructing new polynomial families, derivation of explicit identities, and establishing analytic properties. It opens new avenues for future research by bridging classical and generalized combinatorial sequences within a robust analytic framework.

This paper presents new approximation formulas for the tangent polynomials and Apostol-tangent polynomials of complex order, specifically for large values of n. These polynomials are parameterized by a,b, and c. The derivation of these formulas is accomplished through contour integration techniques, where the contour is carefully selected to avoid branch cuts introduced by the presence of multiple singularities within the integration path. The analysis includes a detailed computation of the singularities associated with the generating functions used in this process, ensuring the accuracy and rigor of the derived formulas. Additionally, the paper provides corollary results that reinforce and affirm the newly established formulas, offering a comprehensive understanding of the behavior of these polynomials under specified conditions.

Core distributions of Maximum Entropy Theory of Ecology (METE) are the Spatial Structure Function (SSF) and the Ecosystem Structure Function (ESF). SSF is a by-species prediction of the clustering of individuals over space. ESF is a kind of container function that describes the probability space of how abundances are assigned to species and how metabolic energy is parti-tioned over individuals in a community. In this study, these core functions of METE are generalized by deriving the corresponding functions in the Tsallis q-entropy. Derivation used the method of Lagrange multipliers. The generalized SSF and ESF are expressed in terms of the q-exponential function. Numerical examples are provided to illustrate the generalized SSF.

In this paper, we unveil a novel category of Frobenius-Genocchi polynomials, grounded in the Bell numbers and Apostol-type functions. Our exploration delves into a comprehensive examination of these polynomials, elucidating various properties. Employing diverse analytical methods and leveraging generating functions for Bell-based Apostol-Frobenius-Type poly-Genocchi polynomials of higher order, we derive explicit and implicit summation formulas, complemented by their symmetric identities.