Related papers: The Ramanujan master theorem and its implications …
We describe applications of the classical umbral calculus to bilinear generating functions for polynomial sequences, identities for Bernoulli and related numbers, and Kummer congruences.
For two arithmetical functions $f$ and $g$, we study the convolution sum of the form $\sum_{n \le N} f(n) g(n+h)$ in the context of its asymptotic formula with explicit error terms. Here we introduce the concept of finite Ramanujan…
In this paper, we study some properties of Euler polynomials arising from umbral calculus. Finally, we give some interesting identities of Euler polynomials using our results. Recently, Dere and Simsek have studied umbral calculus related…
In part one we prove a theorem about the automorphism of solutions to Ramanujan's differential equations. We also investigate possible applications of the result. In part two we prove a similar theorem about the automorphism of solutions to…
Inspired by ideas from umbral calculus and based on the two types of integrals occurring in the defining equations for the gamma and the reciprocal gamma functions, respectively, we develop a multi-variate version of umbral calculus and of…
In this paper, sums represented in (3) are studied. The expressions are derived in terms of Bessel functions of the first and second kinds and their integrals. Further, we point out the integrals can be written as a Meijer G function.
We present a unified algebraic framework utilizing the formal Bell transform to bridge the Dirichlet convolution of arithmetic functions with the combinatorial structure of infinite Euler-type products. By analyzing the logarithmic…
In this present investigation, we introduce the new class R of bi-univalent functions defined by using the Tremblay fractional derivative operator. Additionally, we use the Faber polynomial expansions and Fibonacci numbers to derive bounds…
We extend the technique of using the Trapezoidal Rule for efficient evaluation of the Special Functions of Mathematical Physics given by integral representations. This technique was recently used for Bessel functions, and here we treat…
We approach Riordan arrays and their generalizations via umbral symbolic methods. This new approach allows us to derive fundamental aspects of the theory of Riordan arrays as immediate consequences of the umbral version of the classical…
We introduce and prove evaluations for families of multiple elliptic integrals by solving special types of ordinary and partial differential equations. As an application, we obtain new expressions of Ramanujan-type series of level 4 and…
We obtain integral representations of the $n$-th derivatives of the Bessel functions with respect to the order. The numerical evaluation of these expressions is very efficient using a double exponential integration strategy. Also, from the…
We evaluate definite integrals involving the product of four modified Bessel functions of the first and second kind and a power function. We provide general formulas expressed in terms of the Meijer $G$-function and generalized…
Inspired by the recent pioneering work, dubbed "The Ramanujan Machine" by Raayoni et al. (arXiv:1907.00205), we (automatically) [rigorously] prove some of their conjectures regarding the exact values of some specific infinite continued…
This paper is concerned with the study of the fractional finite sums theory. We present the classes of functions for which it is possible to characterize the constant related to the derivative of fractional sums (denominated by essence of a…
In this paper, we consider several special polynomials related to associated sequences of polynomials. Finally, we give some new and interesting identities of those polynomials arising from transfer formula for the associated sequences.
Based on operator algebras commonly used in quantum mechanics some properties of special functions such as Hermite and Laguerre polynomials and Bessel functions are derived.
Most of the special functions of mathematical physics are connected with the representation of Lie groups. The action of elements $D$ of the associated Lie algebras as linear differential operators gives relations among the functions in a…
We give an elementary proof for new strict upper and lower bounds for the correction term in Ramanujan's approximation for the factorial function
We consider a class of generalized binomials emerging in fractional calculus. After establishing some general properties, we focus on a particular yet relevant case, for which we provide several ready-for-use combinatorial identities,…