Some Properties of the $q$-Vietoris Number Sequence

It is impossible to overplay this function's pervasive role in mathematics. From its humble origin as a complex-valued generalization of the shifted factorial function to its more sophisticated guise as the Mellin transform of e-x, the gamma function surfaces in the study of special functions everywhere. On the other hand it can be argued, following Weierstrass, that the reciprocal of the gamma function is a more natural player, one equally deserving of a central part. In this article we focus on Laplace's basic result and its numerous ramifications. Our first goal is to provide two simple proofs of his identity. We avoid popular techniques: messy contour integration (the kind that deals with branch points) and the theory of Fourier transforms, which invariably invokes the Fourier inversion formula. The only facts about the gamma function that we use follow from Euler's definition. These are its recurrence relation

The objective of present research is to introduce and analyze a new special function that occurs as a part of the kernel of the integral representation of the λ-generalized Hurwitz-Lerch zeta functions. One important aspect of the analysis of special functions is to study their properties. It is found that λ-generalized gamma function satisfies the log convex and derivative properties. Recurrence relation and reflection formula are achieved that are always important to study the behavior of new functions. As an application, a new series representation of the λ-generalized Hurwitz-Lerch zeta functions is established. These new results are validated by analyzing their important cases that agree with the known results. It is interesting to note that the coefficients in the series representation of the family of zeta functions are generalized from “1” to “gamma function” and then to “generalized gamma” and “λ-generalized gamma functions” in a simple and natural way.

A variety of identities involving harmonic numbers and generalized harmonic numbers have been investigated since the distant past and involved in a wide range of diverse fields such as analysis of algorithms in computer science, various branches of number theory, elementary particle physics and theoretical physics. Here we show how one can obtain further interesting identities about certain finite series involving binomial coefficients, harmonic numbers and generalized harmonic numbers by applying the usual differential operator to a known identity. MSC:11M06, 33B15, 33E20, 11M35, 11M41, 40C15.

A variety of identities involving harmonic numbers and generalized harmonic numbers have been investigated since the distant past and involved in a wide range of diverse fields such as analysis of algorithms in computer science, various branches of number theory, elementary particle physics, and theoretical physics. Here we show how one can obtain further interesting and (almost) serendipitous identities about certain finite or infinite series involving binomial coefficients, harmonic numbers, and generalized harmonic numbers by simply applying the usual differential operator to well-known Gauss’s summation formula for2F1(1).

Confluent Appell polynomials

An infinite family of Boolean polynomials which correspond to the discrete average maps, defined in [2], is constructed and their algebraic and combinatorial properties are investigated. They turn out to be balanced, and some recurrence formulas among them are derived.

In this paper, we have determined the limit of ratio of (n+1)th term to the nth term of famous sequences in mathematics like Fibonacci Sequence, Fibonacci – Like Sequence, Pell's Sequence, Generalized Fibonacci Sequence, Padovan Sequence, Generalized Padovan Sequence, Narayana Sequence, Generalized Narayana Sequence, Generalized Recurrence Relations of Fibonacci – Type sequence, Polygonal Numbers, Catalan Sequence, Cayley numbers, Harmonic Numbers and Partition Numbers. We define this ratio as limiting ratio of the corresponding sequence. Sixteen different classes of special sequences are considered in this paper and we have determined the limiting ratios for each one of them. In particular, we have shown that the limiting ratios of Fibonacci sequence and Fibonacci – Like sequence is the fascinating real number called Golden Ratio which is 1.618 approximately. We have shown that the limiting ratio of Pell's sequence is a real number called Silver Ratio and the limiting ratios for generalized Fibonacci sequence are metallic ratios. We have also obtained the limiting ratios of Padovan and generalized Padovan sequence. The limiting ratio of Narayana sequence happens to be a number called super Golden Ratio which is 1.4655 approximately. We have shown that the limiting ratios of Generalized Narayana sequence are the numbers known as super Metallic Ratios. We have also shown that the limiting ratio of generalized recurrence relation of Fibonacci type is 2 and that of Polygonal numbers and Harmonic numbers are 1. We have proved that the limiting ratio of the famous Catalan sequence and Cayley numbers are 4. Finally, assuming Rademacher's Formula, we have shown that the limiting ratio of Partition numbers is the natural logarithmic base e. We have proved fourteen theorems to derive limiting ratios of various well known sequences in this paper. From these limiting ratio values, we can understand the asymptotic behavior of the terms of all these amusing sequences of numbers in mathematics. The limiting ratio values also provide an opportunity to apply in lots of counting and practical problems.

This study presents a detailed investigation into the analytical properties of the generalized Gamma function introduced by Dilcher. We establish a fundamental recurrence relation and derive novel reflection formula for this generalized function, extending the classical identities known for the Euler Gamma function. Further, we analyze the generalized digamma function, the logarithmic derivative of the generalized Gamma function, and obtain a significant difference equation that characterizes its behavior. Explicit expressions for the evaluation of this generalized digamma function at half‐integer arguments are provided. The study also introduces the natural extension to generalized polygamma functions and reports on several special values for both classical and generalized cases. To complement the theoretical analysis, a graphical representation of the generalized digamma function is included, alongside supporting numerical computations that illustrate its properties and validate the derived results. Our findings provide a comprehensive framework that enhances the understanding of these generalized special functions and underscores their potential for broader application in mathematical physics and number theory.

The simultaneous diophantine approximation of values of functions satisfying certain n variable q-difference equations

This paper ensures an extensive survey of the generalization of the various hybrid numbers and hybrid polynomials especially as part of its enhancing importance in the disciplines of mathematics and physics. In this paper, by using the Horadam polynomials, we define the Horadam hybrid polynomials called Horadam hybrinomials. We obtain some special cases and algebraic properties of the Horadam hybrinomials such as recurrence relation, generating function, exponential generating function, Binet formula, summation formulas, Catalan's identity, Cassini's identity and d'Ocagne's identity, respectively. Moreover, we give some applications related to the Horadam hybrinomials in matrices.

We show that isomorphism of effect algebras preserves properties of effect algebras derived from effect algebraic sum ? of elements. These are partial order, order convergence, order topology, existence of states and other important properties. However, there are properties of effect algebras for which the preservation of the ?-operation is not substantial and they need not be preserved.

The Riesz means, or sometimes typical means, were introduced by M. Riesz and have been studied in connection with summability of Fourier series and of Dirichlet series [8] and [11]. In number-theoretic context, it is the Riesz sum rather than the Riesz mean that has been extensively studied. The Riesz sums appear as long as there appears the G-function. Cf. Remark 1 and [14]. As is shown below, the Riesz sum corresponds to integration while Landau's differencing is an analogue of differentiation. This integration-differentiation aspect has been the driving force of many researches on number-theoretic asymptotic formulas. Ingham's decent treatment [13] of the prime number theorem is one typical example. We state some efficient theorems that give asymptotic formulas for the sums of coefficients of the generating Dirichlet series not necessarily satisfying the functional equation.

Ramanujan’s formula for the harmonic number

We consider q-difference equations for colored Jones polynomials. These sequences of polynomials are invariants for the knots and their asymptotics plays an important role in the famous volume conjecture for the complement of the knot to the 3d sphere. We give an introduction to the theory of hyperbolic volume of the knots complements and study the asymptotics of the solutions of q-recurrence relations of high order.

In this paper we express the matrix coefficients of the Fock representation of a q-oscillator algebra in terms of the d-orthogonal Al-Salam Carlitz polynomials. Also, we derive a generating functions, recurrence relations and q-difference equations for these d-orthogonal polynomials.