A new extension of q-Euler numbers and polynomials related to their interpolation functions

AbstractIn this work, by using a p-adic q-Volkenborn integral, we construct a new approach to generating functions of the (h,q)-Euler numbers and polynomials attached to a Dirichlet character χ. By applying the Mellin transformation and a derivative operator to these functions, we define (h,q)-extensions of zeta functions and l-functions, which interpolate (h,q)-extensions of Euler numbers at negative integers

Remarks on Sum of Products of <inline-formula> <graphic file="1029-242X-2008-816129-i1.gif"/></inline-formula>-Twisted Euler Polynomials and Numbers

Abstract The main purpose of this paper is to construct generating functions of higher-order twisted -extension of Euler polynomials and numbers, by using -adic, -deformed fermionic integral on . By applying these generating functions, we prove complete sums of products of the twisted -extension of Euler polynomials and numbers. We also define some identities involving twisted -extension of Euler polynomials and numbers.</p

Multivariate Interpolation Functions of Higher-Order q-Euler Numbers and Their Applications

The aim of this paper, firstly, is to construct generating functions of q-Euler numbers and polynomials of higher order by applying the fermionic p-adic q-Volkenborn integral, secondly, to define multivariate q-Euler zeta function (Barnes-type Hurwitz q-Euler zeta function) and l-function which interpolate these numbers and polynomials at negative integers, respectively. We give relation between Barnes-type Hurwitz q-Euler zeta function and multivariate q-Euler l-function. Moreover, complete sums of products of these numbers and polynomials are found. We give some applications related to these numbers and functions as well

The Effect of Vertex and Edge Removal on Sombor Index

A vertex degree based topological index called the Sombor index was recently defined in 2021 by Gutman and has been very popular amongst chemists and mathematicians. We determine the amount of change of the Sombor index when some elements are removed from a graph. This is done for several graph elements, including a vertex, an edge, a cut vertex, a pendant edge, a pendant path, and a bridge in a simple graph. Also, pendant and non-pendant cases are studied. Using the obtained formulae successively, one can find the Sombor index of a large graph by means of the Sombor indices of smaller graphs that are just graphs obtained after removal of some vertices or edges. Sometimes, using iteration, one can manage to obtain a property of a really large graph in terms of the same property of many other subgraphs. Here, the calculations are made for a pendant and non-pendant vertex, a pendant and non-pendant edge, a pendant path, a bridge, a bridge path from a simple graph, and, finally, for a loop and a multiple edge from a non-simple graph. Using these results, the Sombor index of cyclic graphs and tadpole graphs are obtained. Finally, some Nordhaus–Gaddum type results are obtained for the Sombor index

New families of special numbers and polynomials arising from applications of p-adic q-integrals

Abstract In this manuscript, generating functions are constructed for the new special families of polynomials and numbers using the p-adic q-integral technique. Partial derivative equations, functional equations and other properties of these generating functions are given. With the help of these equations, many interesting and useful identities, relations, and formulas are derived. We also give p-adic q-integral representations of these numbers and polynomials. The results we have obtained for these special numbers and polynomials are closely related to well-known families of polynomials and numbers including the Bernoulli numbers, the Apostol-type Bernoulli numbers and polynomials and the Frobenius-Euler numbers, the Stirling numbers, and the Daehee numbers. We give some remarks and observations on the results of this paper