Some identities involving Appell polynomials

Closed form expressions for Appell polynomials

We show that any Appell sequence can be written in closed form as a forward difference transformation of the identity. Such transformations are actually multipliers in the abelian group of the Appell polynomials endowed with the operation of binomial convolution. As a consequence, we obtain explicit expressions for higher order convolution identities referring to various kinds of Appell polynomials in terms of the Stirling numbers. Applications of the preceding results to generalized Bernoulli and Apostol-Euler polynomials of real order are discussed in detail.

New Convolution Identities for Hypergeometric Bernoulli Polynomials

New convolution identities of hypergeometric Bernoulli polynomials are presented. Two different approaches to proving these identities are discussed, corresponding to the two equivalent definitions of hypergeometric Bernoulli polynomials as Appell sequences.

一个与Andrews-Gordon恒等式相关的新$U$-型函数的混合模拟模性

The mixed mock modularity of a new $U$-type function related to the Andrews-Gordon identities

该研究解决了Bringmann等人关于新向量值$U$-型函数的问题，给出了对应Hecke-Appell型和的混合模拟模形式表达式，该函数源于Andrews-Gordon恒等式相关分拆生成函数的特殊多项式研究。

八阶仿θ函数$V_{0}(q)$的拉马努金型恒等式的简短证明

Short proofs of Ramanujan-like identities for the eighth order mock theta function $V_{0}(q)$

该研究利用Appell函数性质给出八阶仿θ函数$V_0(q)$的拉马努金型恒等式的简短证明，并参照相关经典结果提出恒等式的推广形式。

Umbral calculus and Frobenius-Euler polynomials

In this paper, we study some properties of umbral calculus related to Appell sequence. From those properties, we derive new and interesting identities of Frobenius-Euler polynomials.

Andrews-Beck $NT$ 函数的变换性质与广义 Appell-Lerch 级数

Transformation properties of Andrews-Beck $NT$ functions and generalized Appell-Lerch series

本文加强并扩展了Mao关于 $NT$ 函数变换性质的工作，提供了Hickerson和Mortenson关于秩函数工作的类比，并推导出包含 $NT(r,m,n)$ 及 $M_ω(r,m,n)$ 的恒等式，同时给出了广义 Appell-Lerch 级数的新性质。

On a dual and an overpartition generalization of a family of identities of Andrews

We present a dual of a family of partition identities of Andrews involving partitions with no repeated odd parts (among other conditions), along with an overpartition generalization that encapsulates both families. These were discovered during the course of research for an upcoming article by the authors along with Debajyoti Nandi. The proof uses Appell's comparison theorem.

New identities involving q-Euler polynomials of higher order

In this paper we give new identities involving q-Euler polynomials of higher order.

Identities involving Laguerre polynomials derived from umbral calculus

In this paper, we investigate some identities of Laguerre polynomials involving Bernoulli and Euler polynomials which are derived from umbral calculus.

Some identities involving Gegenbauer polynomials

In this paper we derive some interesting identities arising from the orhtogonality of gegenbauer polynomials.

Some remarkable infinite product identities involving Fibonacci and Lucas numbers

By applying the classic telescoping summation formula and its variants to identities involving inverse hyperbolic tangent functions having inverse powers of the golden ratio as arguments and employing subtle properties of the Fibonacci and Lucas numbers, we derive interesting general infinite product identities involving these numbers.

On the q-binomial identities involving the Legendre symbol modulo 3

I use polynomial analogue of the Jacobi triple product identity together with the Eisenstein formula for the Legendre symbol modulo 3 . to prove six identities involving the $q$-binomial coefficients. These identities are then extended to the new infinite hierarchies of q-series identities by means of the special case of Bailey's lemma. Some of the identities of Ramanujan, Slater, McLaughlin and Sills are obtained this way.

Identities involving values of Bernstein, q-Bernoulli and q-Euler polynomials

In this paper we give some relation involving values of q-Bernoulli, q-Euler and Bernstein polynomials. From these relations, we obtain some interesting identities on the q-Bernoulli, q-Euler and Bernstein polynomials.

Combinatorial Identities Involving Mertens Function Through Relatively Prime Subsets

In this note we give some identities which involve the Mertens function M(n). Our proofs are combinatorial with relatively prime subsets as a main tool.

Fibonacci Identities Involving Reciprocals of Binomial Coefficients

We derive some Fibonacci and Lucas identities which contain inverse binomial coefficients. Extension of the results to the general Horadam sequence is possible, in some cases.

Combinatorial Identities Involving the Mobius Function

In this paper we derive some identities and inequalities on the Möbius mu function. Our main tool is phi functions for intervals of positive integers and their unions.

Some identities involving associated sequences of special polynomials

In this paper, we study some properties of associated sequences of special polynomials. From the properties of associated sequences of polynomials, we derive some interesting identities of special polynomials.

An explanation for the similar appearance of two identities involving Dickson polynomials

This article explains the similar appearance of two polynomial identities involving Dickson polynomials in char. 2, one found by Abhyankar, Cohen, and Zieve, and the other found by the author.

Identities involving Bessel polynomials arising from linear differential equations

In this paper, we study linear differential equations arising from Bessel polynomials and their applications. From these linear differential equations, we give some new and explicit identities for Bessel polynomials.

Some identities for Bernoulli polynomials involving Chebyshev polynomials

In this paper, we derive some new and interesting idebtities for Bernoulli, Euler and Hermite polynomials associated with Chebyshev polynomials.

Two identities involving the cubic partition function

We give a new proof of Chan's identity involving the cubic partition function and we also give a new identity for the cubic partition function which is analogues to the Zuckerman's identity for the ordinary partition function.

涉及与多个退化多项式和数相关的退化Stirling数的若干恒等式

Some identities involving degenerate Stirling numbers associated with several degenerate polynomials and numbers

本文研究了与退化超调和数及退化Bernoulli、Euler、Bell和Fubini多项式相关的两类退化Stirling数的性质、递推关系和恒等式。

涉及双重求和的 Rogers-Ramanujan 型恒等式

Rogers-Ramanujan Type Identities Involving Double Sums

本文利用常数项方法和 Rogers-Szegő 多项式的性质，从经典 Rogers-Ramanujan 恒等式出发，证明了四个新的双重级数 Rogers-Ramanujan 型恒等式。

Identities involving Frobenius-Euler polynomials arising from non-linear differential equations

In this paper we consider non-linear differential equations which are closely related to the generating functions of Frobenius-Euler polynomials. From our non-linear differential equations, we derive some new identities between the sums of products of Frobenius-Euler polynomials and Frobenius-Euler polynomials of higher order.

Some identities involving degenerate r-Stirling numbers

Recently, authors studied the unsigned degenerate r-Stirling number of the first kind and the degenerate r-Stirling number of the second kind, respectively of which are the degenerate versions of the unsigned r-Stirling numbers of the first kind and those of the r-Stirling numbers of the second kind. The aim of this paper is to derive some identities involving such special numbers from the inverse relations for them.

Some identities involving special numbers and moments of random variables

In this paper, we derive some identities involving special numbers and moments of random variables by using the generating functions of the moments of certain random variables. Here the related special numbers are Stirling numbers of the first and second kinds, degenerate Stirling numbers of the first and second kinds, derangement numbers, higher-order Bernoulli numbers and Bernoulli numbers of the second kind.

Identities and congruences involving the Fubini polynomials

In this paper, we investigate the umbral representation of the Fubini polynomials $F_{x}^{n}:=F_{n}(x)$ to derive some properties involving these polynomials. For any prime number $p$ and any polynomial $f$ with integer coefficients, we show $(f(F_{x}))^{p}\equiv f(F_{x})$ and we give other curious congruences.

Identities involving degenerate harmonic and degenerate hyperharmonic numbers

Harmonic numbers have been studied since antiquity, while hyperharmonic numbers were intoduced by Conway and Guy in 1996. The degenerate harmonic numbers and degenerate hyperharmonic numbers are their respective degenerate versions. The aim of this paper is to further investigate some properties, recurrence relations and identities involving the degenerate harmonic and degenerate hyperharmonic numbers in connection with degenerate Stirling numbers of the first kind, degenerate Daehee numbers and degenerate derangements.

New Finite and Infinite Summation Identities Involving the Generalized Harmonic Numbers

We state and prove a general summation identity. The identity is then applied to derive various summation formulas involving the generalized harmonic numbers and related quantities. Interesting results, mostly new, are obtained for both finite and infinite sums. The high points of this paper are perhaps the discovery of several previously unknown infinite summation results involving {\em non-linear} generalized harmonic number terms and the derivation of interesting alternating summation formulas involving these numbers

Congruence identities of regularized multiple zeta values involving a pair of index sets

Riemann zeta values are generalized to multiple zeta values (MZVs) by use of nested sums, and MZVs are generalized to regularized multiple zeta values (RMZVs) by regularization of divergent infinite series. In the present paper, we prove congruence identities of RMZVs of depth $n$ involving a pair of index sets; the congruence relation is given by the vector space spanned by MZVs of depth $n-1$ and products of MZVs. We also obtain a proof of the parity result, and a congruence sum formula for MZVs.

Identities involving cyclic sums of regularized multiple zeta values each of depth less than $5$

In this paper, we give identities involving cyclic sums of regularized multiple zeta values of depth less than $5$. As a corollary, we present two kinds of extensions of Hoffman's theorem for symmetric sums of multiple zeta values for this case.

Generalizations of Fibonacci-Lucas inverse tangent summation identities of Hoggatt and Ruggels

We derive generalizations of a couple of inverse tangent summation identities involving Fibonacci and Lucas numbers. As byproducts we establish many new inverse tangent identities involving the Fibonacci and Lucas numbers.

Nair's and Farhi's identities involving the least common multiple of binomial coefficients are equivalent

In 1982, Nair proved the identity: ${\rm lcm}({k \choose 1}, 2{k \choose 2}, >..., k{k \choose k}) ={\rm lcm}(1, 2, ..., k), \forall k\in \mathbb{N}.$ Recently, Farhi proved a new identity: ${\rm lcm}({k \choose 0}, {k \choose 1}, >..., {k \choose k}) =\frac{{\rm lcm}(1, 2, ..., k+1)}{k+1}, \forall k\in \mathbb{N}.$ In this note, we show that Nair's and Farhi's identities are equivalent.

Some New Identities of Genocchi Numbers and Polynomials involving Bernoulli and Euler polynomials

In this paper, we will deal with some new formulae for two product Genocchi polynomials together with both Euler polynomials and Bernoulli polynomials. We get some applications for Genocchi polynomials. Our applications possess a number of interesting properties to study in Theory of Analytic numbers which we express in the present paper.

Rogers-Ramanujan type identities and Chebyshev Polynomials of the third kind

It is known that $q$-orthogonal polynomials play an important role in the field of $q$-series and special functions. During studying Dyson's "favorite" identity of Rogers--Ramanujan type, Andrews pointed out that the classical orthogonal polynomials also have surprising applications in the world of $q$. By inserting Chebyshev polynomials of the third and the fourth kinds into Bailey pairs, Andrews derived a family of Rogers--Ramanujan type identities and also results related to mock theta functions and Hecke--type series. In this paper, by constructing a new Bailey pair involving Chebyshev pol…

Combinatorial Sums and Identities Involving Generalized Sum-of-Divisors Functions with Bounded Divisors

The class of Lambert series generating functions (LGFs) denoted by $L_α(q)$ formally enumerate the generalized sum-of-divisors functions, $σ_α(n) = \sum_{d|n} d^α$, for all integers $n \geq 1$ and fixed real-valued parameters $α\geq 0$. We prove new formulas expanding the higher-order derivatives of these LGFs. The results we obtain are combined to express new identities expanding the generalized sum-of-divisors functions. These new identities are expanded in the form of sums of polynomially scaled multiples of a related class of divisor sums depending on $n$ and $α$.

Identities Involving Zeros of Ramanujan and Shanks Cubic Polynomials

In this paper we highlight the connection between Ramanujan cubic polynomials (RCPs) and a class of polynomials, the Shanks cubic polynomials (SCPs), which generate cyclic cubic fields. In this way we provide a new characterization for RCPs and we express the zeros of any RCP in explicit form, using trigonometric functions. Moreover, we observe that a cyclic transform of period three permutes these zeros. As a consequence of these results we provide many new and beautiful identities. Finally we connect RCPs to Gaussian periods, finding a new identity, and we study some integer sequences relate…

Some identities involving derangement polynomials and numbers

The problem of counting derangements was initiated by Pierre Remonde de Motmort in 1708. A derangement is a permutation that has no fixed points and the derangement number Dn is the number of fixed point free permutations on an n element set. Furthermore, the derangement polynomials are natural extensions of the derangement numbers. In this paper, we study the derangement polynomials and numbers, their connections with cosine-derangement polynomials and sine-derangement polynomials and their applications to moments of some variants of gamma random variables.

Identities involving q-Genocchi numbers and polynomials

In this paper, we focus on the q-Genocchi numbers and polynomials. We shall introduce new identities of the q-Genocchi numbers and polynomials by using the fermionic p-adic integral on Zp which are very important in the study of Frobenius-Genocchi numbers and polynomials. Also, we give Cauchy-integral formula for the q-Genocchi polynomials and moreover by using measure theory on p-adic integral we derive the distribution formula q-Genocchi polynomials. Finally, we present a new definition of q-Zeta-type function by using Mellin transformation which is the interpolation function of the q-Genocc…

Some identities involving degenerate Stirling numbers arising from normal ordering

In this paper, we derive some identities and recurrence relations for the degenerate Stirling numbers of the first kind and of the second kind which are degenerate versions of the ordinary Stirling numbers of the first kind and of the second kind. They are deduced from the normal oderings of degenerate integral powers of the number operator and their inversions, certain relations of boson operators and from the recurrence relations of the Stirling numbers themselves. Here we note that, while the normal ordering of an integral power of the number operator is expressed with the help of the Stirl…

Automorphic Integrals with Log-polynomial Period Functions and Arithmetical Identities

Building on the works of S. Bochner on equivalence of modular relation with functional equation associated to the Dirichlet series, K. Chandrasekharan and R. Narasimhan obtained new equivalences between the functional equation and some arithmetical identities. Sister Ann M. Heath considered the functional equation in the Hawkins and Knopp context and showed its equivalence to two arithmetical identities associated with entire modular cusp integrals involving rational period functions for the full modular group. In this paper we use techniques of Chandrasekharan and Narasimhan and extend the re…

Some identities on generalized harmonic numbers and generalized harmonic functions

The harmonic numbers and generalized harmonic numbers appear frequently in many diverse areas such as combinatorial problems, many expressions involving special functions in analytic number theory and analysis of algorithms. The aim of this paper is to derive some identities involving generalized harmonic numbers and generalized harmonic functions from the beta functions F(x)= B( x+1, n+1), ( n=0,1,2,..) using elementary methods.

On identities involving the sixth order mock theta functions

We present q-series proofs of four identities involving sixth order mock theta functions from Ramanujan's lost notebook. We also show how Ramanujan's identities can be used to give a quick proof of four sixth order identities of Berndt and Chan.

On Some Generalizations to Floor Function Identities of Ramanujan

We give some generalizations to three identities of Srinivasa Ramanujan involving greatest integer function.

二次型、精确覆盖系统与Theta函数的乘积恒等式

Quadratic Forms, Exact Covering Systems, and Product Identities for Theta Functions

本文建立整二次型与精确覆盖系统的联系，提出结构框架生成无穷多涉及拉马努金theta函数的乘积恒等式，统一解释二十二个Rogers-Ramanujan恒等式，并推导三元二次型相关新恒等式。

On some identities in multiplicative number theory

Using elementary means, we prove several identities involving the Möbius function, generalizing in the multidimensional case well-known formulas coming from convolution arguments.

Identities and estimations involving the least common multiple of strong divisibility sequences

In this paper, we first prove that for any strong divisibility sequences $\boldsymbol{a} = \left(a_n\right)_{n\geq 1}$, we have the identity: $\mathrm{lcm} \left\lbrace \binom{n}{0}_{\bf{a}}, \binom{n}{1}_{\bf{a}},\dots, \binom{n}{n}_{\bf{a}} \right\rbrace = \frac{\mathrm{lcm} \left(a_1,\dots , a_n , a_{n+1}\right)}{a_{n+1}}$ $\left(\forall n \geq 1\right)$, generalizing the identity of Farhi (obtained in 2009 for $a_n=n$). Then, we derive from this one some other interesting identities. Finally, we apply those identities to estimate the least common multiple of the consecutive terms of some L…

New identities involving Bernoulli and Euler polynomials

In this paper we obtain several new identities for Bernoulli and Euler polynomials; some of them extend Miki's and Matiyasevich's identities. Our new method involves differences and derivatives of polynomials.

Koshliakov kernel and identities involving the Riemann zeta function

Some integral identities involving the Riemann zeta function and functions reciprocal in a kernel involving the Bessel functions $J_{z}(x), Y_{z}(x)$ and $K_{z}(x)$ are studied. Interesting special cases of these identities are derived, one of which is connected to a well-known transformation due to Ramanujan, and Guinand.

New identities involving Bernoulli and Euler polynomials. II

We derive several symmetric identities for Bernoulli and Euler polynomials which imply some known identities. Our proofs depend on the new technique developed in part I and some identities obtained in [European J. Combin. 24(2003), 709--718].

Identities involving the $\left(h,q\right)$-Genocchi polynomials and $\left(h,q\right)$-Zeta-type function

The fundamental objective of this paper is to obtain some interesting properties for $\left(h,q\right)$-Genocchi numbers and polynomials by using the fermionic $p$-adic $q$-integral on $\mathbb{Z}_{p}$ and mentioned in the paper $q$-Bernstein polynomials. By considering the $q$-Euler zeta function defined by T. Kim, which can also be obtained by applying the Mellin transformation to the generating function of $\left(h,q\right)$-Genocchi polynomials, we study $\left(h,q\right)$-Zeta-type function. We derive symmetric properties of $\left(h,q\right)$-Zeta function and from these properties we gi…

Elementary Polynomial Identities Involving $q$-Trinomial Coefficients

We use $q$-binomial theorem to prove three new polynomial identities involving $q$-trinomial coefficients. We then use summation formulas for the $q$-trinomial coefficients to convert our identities into another set of three polynomial identities, which imply Capparelli's partition theorems when the degree of the polynomial tends to infinity. This way we also obtain an interesting new result for the sum of the Capparelli's products. We finish this paper by proposing an infinite hierarchy of polynomial identities.

Some Harmonic Number Identities involving certain Reciprocals

Some finite series of harmonic numbers involving certain reciprocals are evaluated. Products of such reciprocals are expanded in a sum of the individual reciprocals, leading to a computer program. A list of examples is provided.

On Rogers-Ramanujan functions, binary quadratic forms and eta-quotients

In a handwritten manuscript published with his lost notebook, Ramanujan stated without proofs forty identities for the Rogers-Ramanujan functions. We observe that the functions that appear in Ramanujan's identities can be obtained from a Hecke action on a certain family of eta products. We establish further Hecke-type relations for these functions involving binary quadratic forms. Our observations enable us to find new identities for the Rogers-Ramanujan functions and also to use such identities in turn to find identities involving binary quadratic forms.