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We prove the following conjecture of Z.-W. Sun ['On congruences related to central binomial coefficients', J. Number Theory 13 (11) (2011), 2219–2238]. Let p be an odd prime. Then $$ \begin{align*} \sum_{k=1}^{p-1}\frac{\binom{2k}k}{k2^k}\equiv-\frac12H_{{(p-1)}/2}+\frac7{16}p^2B_{p-3}\pmod{p^3}, \end{align*} $$ where $H_n$ is the n th harmonic number and $B_n$ is the n th Bernoulli number. In addition, we evaluate $\sum _{k=0}^{p-1}(ak+b)\binom {2k}k/2^k$ modulo $p^3$ for any p -adic integers $a, b$. [ABSTRACT FROM AUTHOR]

In this study, we use discrete Appell convolution to define the sequence of Laguerre type twice-iterated Appell polynomials. We obtain explicit representation, recurrence relation, determinantal representation, lowering operator, integro-partial raising operator and integro-partial differential equation. In addition, the special cases of this new family are investigated using Euler and Bernoulli numbers. We also state their corresponding characteristic properties. [ABSTRACT FROM AUTHOR]

This article, published in Mathematical Notes, discusses lacunary recurrence relations with gaps of length 8 for the Bernoulli and Euler polynomials. The authors use standard notations for these polynomials and numbers and provide formulas and expansions for them. They also mention the history of lacunary recurrence relations for these numbers and polynomials and highlight their own contributions in finding such relations. The authors describe the method of proof for their main result, Theorem 1, and provide corollaries that extend the relations to the Bernoulli and Euler numbers. The article concludes by acknowledging funding support and declaring no conflicts of interest. [Extracted from the article]

We extended the classical Bernoulli and Euler numbers and polynomials to introduce the Laguerre-type Bernoulli and Euler numbers and related fractional polynomials. The case of fractional Bernoulli and Euler polynomials and numbers has already been considered in a previous paper of which this article is a further generalization. Furthermore, we exploited the Laguerre-type fractional exponentials to define a generalized form of the classical Laplace transform. We show some examples of these generalized mathematical entities, which were derived using the computer algebra system Mathematica© (latest v. 14.0). [ABSTRACT FROM AUTHOR]

The paper has two main objectives. Firstly, it explores the properties of hyperbolic cosine and hyperbolic sine functions by using Volkenborn and the fermionic p-adic integrals, respectively. It also investigates interesting identities involving Bernoulli and degenerate Bernoulli numbers from differential equations with their solutions. Secondly, it introduces a random variable whose mass function is given in terms of Euler polynomials and find some expressions for the expectation of the random variable. [ABSTRACT FROM AUTHOR]

The aim of this paper is to introduce several degenerate hyperbolic functions as degenerate versions of the hyperbolic functions, to evaluate Volkenborn and the fermionic p-adic integrals of the degenerate hyperbolic cosine and the degenerate hyperbolic sine functions and to derive from them some identities involving the degenerate Bernoulli numbers, the degenerate Euler numbers and the Cauchy numbers of the first kind. [ABSTRACT FROM AUTHOR]

The restricted partition function p N (n) counts the number of partitions of n into at most N parts. In the nineteenth century, Sylvester showed that these partition functions can be expressed as a sum of k-periodic quasi-polynomials ( 1 ≤ k ≤ N ) which he termed as Waves. It is now well-known that one can easily perform a wave decomposition using a special type of partial fraction decomposition (the so-called q-partial fractions) of the generating function of p N (n) . In this paper, we show that the coefficients of these q-partial fractions can be expressed as a linear combination of the Ramanujan sums. In particular, we show, for the first time, an appearance of the Degenerate Bernoulli numbers, the Degenerate Euler numbers and a special generalization of the Ramanujan sums, which we term as a Gaussian–Ramanujan sum, in the formulae for certain waves. These coefficients not only provide a good approximation of p N (n) but they can also be used for obtaining good bounds. Further, we provide a combinatorial meaning to these sums. Our approach for partial fractions is based on a projection operator on the I-adic completion of the ring of polynomials, where I is an ideal generated by the cyclotomic polynomial. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to construct generating functions in terms of hypergeometric function and logarithm function for finite and infinite sums involving higher powers of inverse binomial coefficients. These generating functions provide a novel way of examining higher powers of inverse binomial coefficients from the perspective of these sums, assessing how several of these sums and these coefficients are related to each other. A relation between the Euler–Frobenius polynomial and B‐spline associated with exponential Euler spline is reported. Moreover, with the aid of derivative operator and functional equations for generating functions, many new computational formulas involving the special finite sums of higher powers of (inverse) binomial coefficients, the Bernoulli polynomials and numbers, the Euler polynomials and numbers, the Stirling numbers, the harmonic numbers, and special finite sums are derived. Moreover, a few recurrence relations containing these particular finite sums are given. Using these recurrence relations, we give a solution of the problem which was given by Charalambides. We give calculations algorithms for these finite sums. Applying these algorithms and Wolfram Mathematica 12.0, we give some plots and many values of these polynomials and finite sums. [ABSTRACT FROM AUTHOR]

Several trigonometric sums are examined by means of partial fraction decompositions. Their generating functions will be evaluated in closed forms and the related asymptotic formulae in terms of Riemann zeta function will be established. [ABSTRACT FROM AUTHOR]

We evaluate in closed form a large class of power series where the coefficients are products of special numbers, or products of special numbers and Laguerre polynomials. A number of examples is presented involving harmonic, hyperharmonic, Fibonacci, Lucas, Bernoulli, Euler, and derangement numbers. We also obtain a new convolution formula for the Laguerre polynomials. [ABSTRACT FROM AUTHOR]

Reciprocal series are employed to systematically review convolution sums, orthogonality relations, recurrence relations and reciprocal formulae for several classical number sequences, such as binomial coefficients, Stirling numbers, Bernoulli numbers, and Euler numbers. [ABSTRACT FROM AUTHOR]

In this paper, we mainly prove the following conjecture of Sun[Congruences involving binomial coefficients and Apery-like numbers, Publ. Math. Debrecen 96 (2020), pp. 315–346]: Let p>3 be a prime. Then ∑ k = 0 p − 1 ( 2 k k ) 3 k + 1 (− 16) k f k ≡ (− 1) (p − 1) / 2 p + p 3 E p − 3 (mod p 4) , where f n = ∑ k = 0 n ( n k ) 3 and E n stand for the nth Franel number and nth Euler number, respectively. [ABSTRACT FROM AUTHOR]

We obtain lacunary recurrent relations with gaps of length four for the Bernoulli and Euler polynomials and, as a consequence, known and new lacunary relations for the Bernoulli and Euler numbers. [ABSTRACT FROM AUTHOR]

The main objective in this paper is to study on special numbers and polynomials that contain finite sums of powers of binomial coefficients. By using generating function methods, some formulas and relations related to these numbers and the Apostol-Bernoulli and Apostol-Euler numbers of negative higher order, the Bernoulli and Euler numbers, the Stirling type numbers, the combinatorial numbers, the Bell polynomials, the Fubini type polynomials, and the Legendre polynomials are presented. Moreover, asymptotic expressions of the finite sums of powers of binomial coefficients for these numbers are given. Some numeric values of these asymptotic expressions are illustrated by the tables. Finally, some inequalities for these numbers are given. [ABSTRACT FROM AUTHOR]

In this paper, we confirm six conjectures on the exact values of some permanents, relating them to the Genocchi numbers of the first and second kinds as well as the Euler numbers. For example, we prove that per [ ⌊ 2 j − k n ⌋ ] 1 ≤ j , k ≤ n = 2 (2 n + 1 − 1) B n + 1 , where B 0 , B 1 , B 2 , ... are the Bernoulli numbers. We also show that per [ sgn (cos ⁡ π i + j n + 1) ] 1 ≤ i , j ≤ n = { − ∑ k = 0 m ( m k ) E 2 k + 1 if n = 2 m + 1 , ∑ k = 0 m ( m k ) E 2 k if n = 2 m , where sgn (x) is the sign function, and E 0 , E 1 , E 2 , ... are the Euler (zigzag) numbers. In the course of linking the evaluation of these permanents to the aforementioned combinatorial sequences, the classical permutation statistic – the excedance number, together with several kinds of its variants, plays a central role. Our approach features recurrence relations, bijections, as well as certain elementary operations on matrices that preserve their permanents. Moreover, our proof of the second permanent identity leads to a proof of Bala's conjectural continued fraction formula, and an unexpected permutation interpretation for the γ -coefficients of the 2-Eulerian polynomials. [ABSTRACT FROM AUTHOR]

In this paper, we use two different approaches to introduce q-analogs of Riemann's zeta function and prove that their values at even integers are related to the q-Bernoulli and q-Euler's numbers introduced by Ismail and Mansour [Analysis and Applications, 17(6) (2019), 853–895]. [ABSTRACT FROM AUTHOR]

The aim of this paper is to give generating functions for parametrically generalized polynomials that are related to the combinatorial numbers, the Bernoulli polynomials and numbers, the Euler polynomials and numbers, the cosine-Bernoulli polynomials, the sine-Bernoulli polynomials, the cosine-Euler polynomials, and the sine-Euler polynomials. We investigate some properties of these generating functions. By applying Euler's formula to these generating functions, we derive many new and interesting formulas and relations related to these special polynomials and numbers mentioned as above. Some special cases of the results obtained in this article are examined. With this special case, detailed comments and comparisons with previously available results are also provided. Furthermore, we come up with open questions about interpolation functions for these polynomials. The main results of this paper highlight the existing symmetry between numbers and polynomials in a more general framework. These include Bernouilli, Euler, and Catalan polynomials. [ABSTRACT FROM AUTHOR]

We evaluate the Hankel determinants of various sequences related to Bernoulli and Euler numbers and special values of the corresponding polynomials. Some of these results arise as special cases of Hankel determinants of certain sums and differences of Bernoulli and Euler polynomials, while others are consequences of a method that uses the derivatives of Bernoulli and Euler polynomials. We also obtain Hankel determinants for sequences of sums and differences of powers and for generalized Bernoulli polynomials belonging to certain Dirichlet characters with small conductors. Finally, we collect and organize Hankel determinant identities for numerous sequences, both new and known, containing Bernoulli and Euler numbers and polynomials. [ABSTRACT FROM AUTHOR]

By using the calculus of finite differences methods and the umbral calculus, we construct recurrence relations for a new class of special numbers. Using this recurrence relation, we define generating functions for this class of special numbers and also new classes of special polynomials. We investigate some properties of these generating functions. By using these generating functions with their functional equations, we obtain many new and interesting identities and relations related to these classes of special numbers and polynomials, the Bernoulli numbers and polynomials, the Euler numbers and polynomials, the Stirling numbers. Finally, some derivative formulas and integral formulas for these classes of special numbers and polynomials are given. In general, this article includes results that have the potential to be used in areas such as discrete mathematics, combinatorics analysis and their applications. [ABSTRACT FROM AUTHOR]

The main purpose of this paper is to obtain computational identities and formulas for a certain class of combinatorial-type numbers and polynomials. By the aid of the generating function technique, we derive a recurrence relation and an infinite series involving the aforementioned class of combinatorial-type numbers. By applying the Riemann integral to the combinatorial-type polynomials with multivariables, we present some integral formulas for these polynomials, including the Bernoulli numbers of the second kind. By the implementation of the p-adic integral approach to the combinatorial-type polynomials with multivariables, we also obtain formulas for the Volkenborn integral and the fermionic p-adic integral of these polynomials. Furthermore, we provide an approximation for the combinatorial-type numbers with the aid of the Stirling's approximation for factorials. By coding some of our results in Mathematica using the Wolfram programming language, we also provide some numerical evaluations and illustrations on the combinatorialtype numbers and their Stirling's approximation with table and figures. We also give some remarks and observations on the combinatorial-type numbers together with their relationships to other well-known special numbers and polynomials. As a result of these observations, we derive some computation formulas containing the Dirichlet series involving the Möbius function, the Bernoulli numbers, the Catalan numbers, the Stirling numbers, the Apostol--Bernoulli numbers, the Apostol--Euler numbers, the Apostol--Genocchi numbers and some kinds of combinatorial numbers. Besides, some inequalities for the combinatorial-type numbers are presented. Finally, we conclude this paper by briefly overviewing the results with their potential applications. [ABSTRACT FROM AUTHOR]

We establish some new combinatorial identities involving Euler polynomials and balancing (Lucas-balancing) polynomials. The derivations use elementary techniques and are based on functional equations for the respective generating functions. From these polynomial relations, we deduce interesting identities with Fibonacci and Lucas numbers, and Euler numbers. The results must be regarded as companion results to some Fibonacci-Bernoulli identities, which we derived in our previous paper. [ABSTRACT FROM AUTHOR]

By means of the generating function approach, we derive several summation formulae involving multiple harmonic numbers H ,,,/■(a), as well as other combinatorial numbers named after Bernoulli, Euler, Bell, Genocchi and Stirling. [ABSTRACT FROM AUTHOR]

We introduce two sine and cosine types of generating functions in a general case and apply them to the generating functions of classical hypergeometric orthogonal polynomials as well as some widely investigated combinatorial numbers such as Bernoulli, Euler and Genocchi numbers. This approach can also be applied to other celebrated sequences. [ABSTRACT FROM AUTHOR]

By introducing a kernel involving an exponent function with multiple parameters, we establish a new Hilbert-type inequality and its equivalent Hardy form. We also prove that the constant factors of the obtained inequalities are the best possible. Furthermore, by introducing the Bernoulli number, Euler number, and the partial fraction expansion of cotangent function and cosecant function, we get some special and interesting cases of the newly obtained inequality. [ABSTRACT FROM AUTHOR]

In the paper, the authors view some ordinary differential equations and their solutions from the angle of (generalized) derivative polynomials and simplify some known identities for the Bernoulli numbers and polynomials, the Frobenius–Euler polynomials, the Euler numbers and polynomials, in terms of the Stirling numbers of the first and second kinds [ABSTRACT FROM AUTHOR]

Most identities of Genocchi numbers and polynomials are related to the well-known Benoulli and Euler polynomials. In this paper, multi poly-Genocchi polynomials with parameters a, b and c are defined by means of polylogarithm in multiple paramaters. Several properties of these polynomials are established including some recurrence relations and explicit formulas. [ABSTRACT FROM AUTHOR]

In [11], Jolany et al. defined generalizations of Apostol-Genocchi numbers and poly- nomials. Most identities on classical or generalized Apostol-Genocchi numbers and polynomials are related to the well-known Bernoulli and Euler numbers and polynomials. However, in this paper, identities on generalized Apostol-Genocchi numbers and polynomials which are not associ- ated with the Bernoulli- and Euler-types are introduced. Specifically, identities involving binomial coefficients and some integral identities which only relate generalized Apostol-Genocchi numbers and polynomials are established. [ABSTRACT FROM AUTHOR]

We introduce poly-Cauchy numbers with level 2. Poly-Cauchy numbers may be interpreted as a kind of generalizations of the classical Cauchy numbers by using the inverse relation of exponentials and logarithms. On the contrary, poly-Bernoulli numbers can be from the inverse relation of logarithms and exponentials. In this similar stream, poly-Cauchy numbers with level 2 may be yielded from the inverse relation about the hyperbolic sine function, which is a 2-step function of the exponential function. In this article, we show several expressions, relations, and properties about poly-Cauchy numbers with level 2. Poly-Cauchy numbers with level 2 can be expressed in terms of multinomial coefficients with combinatorial summation, Stirling numbers of the first kind, or iterated integrals. We also give some recurrence relations for poly-Cauchy numbers with level 2. When the index is negative, the double summation may be formulated as a closed form. A simple case of Cauchy numbers with level 2 has some more relations with D numbers from higher-order Bernoulli numbers or complementary Euler numbers. We prove some more expressions in determinants, continued fractions or by Trudi's formula. [ABSTRACT FROM AUTHOR]

For a positive integer N, hypergeometric Cauchy numbers cN,n are defined by ... where 2F1(a,b;c;z) is the Gauss hypergeometric function. When N = 1, cn = c1,n are the classical Cauchy numbers. In 1875, Glaisher gave several interesting determinant expressions of numbers, including Bernoulli, Cauchy and Euler numbers (see (4), (6) and (3) in the text). Hypergeometric numbers can be recognized as one of the most natural extensions of the classical Cauchy numbers in terms of determinants (see Section 2), though many kinds of generalizations of the Cauchy numbers have been considered by many authors. In addition, there are some relations between the hypergeometric Cauchy numbers and the classical Cauchy numbers. In this paper, we give the determinant expressions of hypergeometric Cauchy numbers and their generalizations, and show some interesting expressions of hypergeometric Cauchy numbers. As applications, we can get the inversion relations such that hypergeometric Cauchy numbers as cN,n/n! and the numbers N/(N + n) are interchanged in terms of determinants of the so-called Hassenberg matrices. [ABSTRACT FROM AUTHOR]

Four classes of multiple series, which can be considered as multifold counterparts of four classical summation formulae related to the Riemann zeta series, are evaluated in closed form. [ABSTRACT FROM AUTHOR]

Two classes of trigonometric sums about integer powers of secant function are evaluated that are closely related to Jordan's totient function. [ABSTRACT FROM AUTHOR]

We introduce and study a variant of Kummer's notion of (ir)regularity of primes which we call G-(ir)regularity and is based on Genocchi rather than Bernoulli numbers. We say that an odd prime p is G-irregular if it divides at least one of the Genocchi numbers G 2 , G 4 , ... , G p − 3 , and G-regular otherwise. We show that, as in Kummer's case, G-irregularity is related to the divisibility of some class number. Furthermore, we obtain some results on the distribution of G-irregular primes. In particular, we show that each primitive residue class contains infinitely many G-irregular primes and establish non-trivial lower bounds for their number up to a given bound x as x tends to infinity. As a byproduct, we obtain some results on the distribution of primes in arithmetic progressions with a prescribed near-primitive root. [ABSTRACT FROM AUTHOR]

In this paper, we make use of the probabilistic method to calculate the moment rep-resentation of generalized higher-order Genocchi polynomials. We obtain the moment expression of the generalized higher-order Genocchi numbers with a and b parameters. Some characteriza-tions and identities of generalized higher-order Genocchi polynomials are given by the proof of the moment expression. As far as properties given by predecessors are concerned, we prove them by the probabilistic method. Finally, new identities of relationships involving generalized higher-order Genocchi numbers and harmonic numbers, derangement numbers, Fibonacci numbers, Bell num-bers, Bernoulli numbers, Euler numbers, Cauchy numbers and Stirling numbers of the second kind are established. [ABSTRACT FROM AUTHOR]

We present a number of identities in the form of Stirling transforms. These identities involve Stirling numbers of the first and second kinds, hyperharmonic and derangement numbers, Bernoulli and Euler numbers and polynomials, powers, power sums, and factorials. [ABSTRACT FROM AUTHOR]

By means of the symmetric summation theorem on polynomial differences due to Chu and Magli [Summation formulae on reciprocal sequences. European J Combin. 2007;28(3):921-930], we examine Bernoulli and Euler polynomials of higher order. Several reciprocal relations on Bernoulli and Euler numbers and polynomials are established, including some recent ones obtained by Agoh Shortened recurrence relations for generalized Bernoulli numbers and polynomials. J Number Theory. 2017;176:149-173. [ABSTRACT FROM AUTHOR]

We prove some lacunary convergent series representations for ζ(n). For example, we prove that ζ(7)=∑k=2∞(−1)k(2π)6k(6k+3)!⋅B6k−6k−1+π69451+12+⋯+19−log(2π), where Bn are the nth Bernoulli numbers. [ABSTRACT FROM AUTHOR]

We define a truncated Euler polynomial Em,n(x) as a generalization of the classical Euler polynomial En(x). In this paper we give its some properties and relations with the hypergeometric Bernoulli polynomial. [ABSTRACT FROM AUTHOR]

The main purpose of this paper is to construct new families of special num- bers with their generating functions. These numbers are related to many well-known numbers, which are Bernoulli numbers, Fibonacci numbers, Lucas numbers, Stirling numbers of the second kind and central factorial numbers. Our other inspiration of this paper is related to the Golombek's problem [15] "Aufgabe 1088. El. Math., 49 (1994), 126-127". Our first numbers are not only related to the Golombek's problem, but also computation of the nega- tive order Euler numbers. We compute a few values of the numbers which are given by tables. We give some applications in probability and statistics. That is, special values of mathematical expectation of the binomial distribu- tion and the Bernstein polynomials give us the value of our numbers. Taking derivative of our generating functions, we give partial differential equations and also functional equations. By using these equations, we derive recurrence relations and some formulas of our numbers. Moreover, we come up with a conjecture with two open questions related to our new numbers. We give two algorithms for computation of our numbers. We also give some combinato- rial applications, further remarks on our new numbers and their generating functions. [ABSTRACT FROM AUTHOR]

The Riemann zeta identity at even integers of Lettington, along with his other Bernoulli and zeta relations, is generalized. Other corresponding recurrences and determinant relations are illustrated. Another consequence is the application to sums of double zeta values. A set of identities for the Ramanujan and generalized Ramanujan polynomials is presented. An alternative proof of Lettington's identity is provided, together with its generalizations to the Hurwitz and Lerch zeta functions, hence to Dirichlet L series, to Eisenstein series, and to general Mellin transforms. The Hurwitz numbers H ˜ n occur in the Laurent expansion about the origin of a certain Weierstrass ℘ function for a square lattice, and are highly analogous to the Bernoulli numbers. An integral representation of the Laurent coefficients about the origin for general ℘ functions, and for these numbers in particular, is presented. As a Corollary, the asymptotic form of the Hurwitz numbers is determined. In addition, a series representation of the Hurwitz numbers is given, as well as a new recurrence. Other results concern the Matter numbers of the equianharmonic case of the ℘ function. [ABSTRACT FROM AUTHOR]

For positive integers N and M, the general hypergeometric Cauchy polynomials c ( z) ( M, N ≥ 1; n ≥ 0) are defined by where $${{}_2 F_1(a,b;c;z)}$$ is the Gauss hypergeometric function. When M = N = 1, c = c are the classical Cauchy numbers. In 1875, Glaisher gave several interesting determinant expressions of numbers, including Bernoulli, Cauchy and Euler numbers. In the aspect of determinant expressions, hypergeometric Cauchy numbers are the natural extension of the classical Cauchy numbers, though many kinds of generalizations of the Cauchy numbers have been considered by many authors. In this paper, we show some interesting expressions of generalized hypergeometric Cauchy numbers. We also give a convolution identity for generalized hypergeometric Cauchy polynomials. [ABSTRACT FROM AUTHOR]

In this paper, we consider some ordinary differential equations associated with modifiled degenerate Euler and Bernoulli numbers and give some new identities for these numbers arising from our differential equations. [ABSTRACT FROM AUTHOR]

A sequence inverse relationship can be defined by a pair of infinite inverse matrices. If those two matrices are the same, they define a dual relationship. Presented here is a unified approach to construct dual relationships using pseudo-Riordan involutions. Then we give four dual relationships for the Bernoulli numbers and the Euler numbers, from which the corresponding dual sequences of the Bernoulli polynomials and the Euler polynomials are constructed. Some applications in the construction of identities of the Bernoulli numbers and polynomials and the Euler numbers and polynomials are discussed based on the dual relationships. [ABSTRACT FROM AUTHOR]

Poly-Euler numbers are introduced as a generalization of the Euler numbers in a manner similar to the introduction of the poly-Bernoulli numbers. In this paper, some number-theoretic properties of poly-Euler numbers, for example, explicit formulas, a Clausen–von Staudt type formula, congruence relations and duality formulas, are given together with their combinatorial properties. [ABSTRACT FROM AUTHOR]

In this paper, we continue to investigate the properties of those sequences satisfying the condition . As applications we deduce some recurrence relations and congruences for Bernoulli and Euler numbers. [ABSTRACT FROM AUTHOR]

Let be a prime, and let be a rational -adic integer. Let and denote the Bernoulli numbers and Bernoulli polynomials given by and . In this paper, we show that where and is given by . [ABSTRACT FROM AUTHOR]