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Let $ \{S_n\} $ be the Apéry-like sequence given by $ S_n = \sum_{k = 0}^n\binom nk\binom{2k}k\binom{2n-2k}{n-k} $. We show that for any odd prime $ p $, $ \sum_{n = 1}^{p-1}\frac {nS_n}{8^n}{\equiv} (1-(-1)^{\frac{p-1}2})p^2\ (\text{ mod}\ {p^3}) $. Let $ \{Q_n\} $ be the Apéry-like sequence given by $ Q_n = \sum_{k = 0}^n\binom nk(-8)^{n-k}\sum_{r = 0}^k\binom kr^3 $. We establish many congruences concerning $ Q_n $. For an odd prime $ p $, we also deduce congruences for $ \sum_{k = 0}^{p-1}\binom{2k}k^3\frac 1{64^k}\ (\text{ mod}\ {p^3}) $, $ \sum_{k = 0}^{p-1}\binom{2k}k^3\frac 1{64^k(k+1)^2}\ (\text{ mod}\ {p^2}) $ and $ \sum_{k = 0}^{p-1}\binom{2k}k^3\frac 1{64^k(2k-1)}\ (\text{ mod}\ p) $, and pose lots of conjectures on congruences involving binomial coefficients and Apéry-like numbers.

In this paper we try to prove that ζ(2n+1) is irrational for all natural number n.Also,in our attempt, we construct an upper bound tothe Zeta values at odd integers.It is interesting to see how the irrational-ity of Zeta values at even positive integers mixed up with Dirichletsirrationality criterion and this bound accelerates our proof further,caseby case.

An important component of Apery’s proof that ζ(3) is irrational involves representing ζ(3) as the limit of the quotient of two rational solutions to a three-term recurrence. We present various appr...

In this paper, we use the methods of contour integration and generating function involving Fuss-Catalan numbers to study some Euler-Apery-type series. In particular, we obtain some explicit formulas for some Euler-Apery-type series. Based on these formulas, we further show that some series are reducible to logarithms (such as: $$\log (2),\log (3),\log (5)$$ etc.), zeta values and multiple polylogarithms. Moreover, we establish a recurrence relation for general Euler-Apery-type series involving multiple harmonic star sum. Furthermore, some interesting new consequences and illustrative examples are considered.

In this paper, we formally introduce the notion of Apery-like sums and we show that every multiple zeta values can be expressed as a Z-linear combination of them. We even describe a natural way to do so. This allows us to put in a new theoretical context several identities scattered in the literature, as well as to discover many new interesting ones. We give in this paper new integral formulas for multiple zeta values and Apery-like sums. They enable us to give a short direct proof of Zagier's formulas for ζ ( 2 , … , 2 , 3 , 2 , … , 2 ) as well as of similar ones in the context of Apery-like sums. The relations between Apery-like sums themselves still remain rather mysterious, but we get significant results and state some conjectures about their pattern.

The Hurwitz zeta function ζ ( s , a ) admits a well-known (divergent) asymptotic expansion in powers of 1 / a involving the Bernoulli numbers. Using Wilson orthogonal polynomials, we determine an effective bound for the error made when this asymptotic series is replaced by nearly diagonal Pade approximants. By specialization, we obtain new fast converging sequences of rational approximations to the values of the Riemann zeta function at every integers ≥2. The latter can be viewed, in a certain sense, as analogues of Apery's celebrated sequences of rational approximations to ζ ( 2 ) and ζ ( 3 ) .

The purpose for this research was to investigate the Riemann zeta function at odd integer values, because there was no simple representation for these results. The research resulted in the closed form expression for representing the zeta function at the odd integer values 2n+1 for n a positive integer. The above representation shows the zeta function at odd positive integers can be represented in terms of the Euler numbers E2n and the polygamma functions ψ(2n)(3/4). This is a new result for this study area. For completeness, this paper presents a review of selected properties of the Riemann zeta function together with how these properties are derived. This paper will summarize how to evaluate zeta (n) for all integers n different from 1. Also as a result of this research, one can obtain a closed form expression for the Dirichlet beta series evaluated at positive even integers. The results presented enable one to construct closed form expressions for the Dirichlet eta, lambda and beta series evaluated at odd and even integers. Closed form expressions for Apery’s constant zeta (3) and Catalan’s constant beta (2) are also presented.

Various methods are used to investigate sums involving a reciprocal central binomial coefficient and a power term. In the first part, new functions are introduced for calculation of sums with a negative exponent in the power term. A recurrence equation for the functions provides an integral representation of the sums using polylogarithm functions. Thus polylogarithms and, in particular, zeta values can be expressed via these functions, too. In the second part, a straightforward recurrence formula is derived for sums having a positive exponent in the power term. Finally, two interesting cases of double sums are presented.

A four-dimensional integral containing g(x,y,z,t)Cn(λ)(x) is derived. Cn(λ)(x) is the Gegenbauer polynomial, g(x,y,z,t) is a product of the generalized logarithm quotient functions and the integral is taken over the region 0≤x≤1,0≤y≤1,0≤z≤1,0≤t≤1. The integral is difficult to compute in general. Special cases are given and invariant index forms are derived. The zero distribution of almost all Hurwitz–Lerch zeta functions is asymmetrical. All the results in this work are new.

The objective of the present paper is to obtain a quadruple infinite integral. This integral involves the product of the Struve and parabolic cylinder functions and expresses it in terms of the Hurwitz–Lerch Zeta function. Almost all Hurwitz-Lerch Zeta functions have an asymmetrical zero distributionSpecial cases in terms fundamental constants and other special functions are produced. All the results in the work are new.

In this paper, we have derived and evaluated a quadruple integral whose kernel involves the logarithm and product of Bessel functions of the first kind. A new quadruple integral representation of Catalan’s G and Apéry’s ζ(3) constants are produced. Some special cases of the result in terms of fundamental constants are evaluated. All the results in this work are new.

This article, dedicated with admiration in memory of Jon and Peter Borwein, illustrates by example, the power of experimental mathematics, so dear to them both, by experimenting with so-called Apery limits and WZ pairs. In particular we prove a weaker form of an intriguing conjecture of Marc Chamberland and Armin Straub (in an article dedicated to Jon Borwein), and generate lots of new Apery limits. We also rediscovered an infinite family of cubic irrationalities, that suggested very good effective irrationality measures (lower than Liouville's generic 3), and that seemed to go down to the optimal 2. It turned out this follows from known deep results in number theory, and a postscript by Paul Voutier outlines the proof. Nevertheless we believe that further experiments with our Maple packages will lead to new and interesting results., 11 pages. Accompanied by three Maple packages and numerous output files available from https://sites.math.rutgers.edu/~zeilberg/mamarim/mamarimhtml/wzp.html ; In fond memory of Jon and Peter Borwein. This version contains a postscript by Paul Voutier

Let \begin{document}$ n $\end{document} be a nonnegative integer. The \begin{document}$ n $\end{document} -th Apery number is defined by \begin{document}$ A_n: = \sum\limits_{k = 0}^n\binom{n+k}{k}^2\binom{n}{k}^2. $\end{document} Z.-W. Sun investigated the congruence properties of Apery numbers and posed some conjectures. For example, Sun conjectured that for any prime \begin{document}$ p\geq7 $\end{document} \begin{document}$ \sum\limits_{k = 0}^{p-1}(2k+1)A_k\equiv p-\frac{7}{2}p^2H_{p-1}\pmod{p^6} $\end{document} and for any prime \begin{document}$ p\geq5 $\end{document} \begin{document}$ \sum\limits_{k = 0}^{p-1}(2k+1)^3A_k\equiv p^3+4p^4H_{p-1}+\frac{6}{5}p^8B_{p-5}\pmod{p^9}, $\end{document} where \begin{document}$ H_n = \sum_{k = 1}^n1/k $\end{document} denotes the \begin{document}$ n $\end{document} -th harmonic number and \begin{document}$ B_0, B_1, \ldots $\end{document} are the well-known Bernoulli numbers. In this paper we shall confirm these two conjectures.

In this paper, we establish some congruences involving the Apéry numbers [Formula: see text]. For example, we show that [Formula: see text] for any positive integer [Formula: see text], and [Formula: see text] for any prime [Formula: see text], where [Formula: see text] is the [Formula: see text]th Bernoulli number. We also present certain relations between congruence properties of the two kinds of Aṕery numbers, [Formula: see text] and [Formula: see text].

We prove a general structure theorem by which an inside factorial monoid can be uniquely decomposed into a factorial monoid and the related Apery set. The structure of the latter is analyzed subsequently. As an example, we study the Cale monoid of nonnegative solutions of equations of the form a 1 x 1 + ⋯ + a r − 1 x r − 1 = a r x r for positive integers a 1 , … , a r . In this case, the monoid is isomorphic to a simplicial full affine semigroup, and the factorial monoid of the decomposition is generated by extremal rays. For the special case r = 3 , we discuss the relationship to a kind of unique representation as studied by Elliott for a r ≤ 10 .

We prove two supercongruences for sums of the Apery-like numbers: $$\begin{aligned} T_n=\sum _{k=0}^n{n\atopwithdelims ()k}^2{2k\atopwithdelims ()n}^2, \end{aligned}$$ which was first introduced by Almkvist and Zudilin. These results confirm two conjectural supercongruences due to Sun. Our proof relies on symbolic summation method and Sun’s transformation formula.

We present explicit formulas for the following family of parametric binomial sums involving harmonic numbers for $$p=0,1,2$$ and $$|t|\le 1$$ . $$\begin{aligned} \sum _{k=1}^{\infty }\frac{H_{k-1}t^k}{k^p\left( {\begin{array}{c}n+k\\ k\end{array}}\right) } \quad \text{ and }\quad \sum _{k=1}^{\infty }\frac{t^k}{k^p\left( {\begin{array}{c}n+k\\ k\end{array}}\right) }. \end{aligned}$$ We also generalize the following relation between the Stirling numbers of the first kind and the Riemann zeta function to polygamma function and give some applications. $$\begin{aligned} \zeta (n+1)=\sum _{k=n}^{\infty }\frac{s(k,n)}{kk!}, \quad n=1,2,3,\ldots . \end{aligned}$$ As examples, $$\begin{aligned} \zeta (3)=\frac{1}{7}\sum _{k=1}^{\infty }\frac{H_{k-1}4^k}{k^2\left( {\begin{array}{c}2k\\ k\end{array}}\right) }, \quad \text{ and }\quad \zeta (3)=\frac{8}{7}+\frac{1}{7}\sum _{k=1}^{\infty } \frac{H_{k-1}4^k}{k^2(2k+1)\left( {\begin{array}{c}2k\\ k\end{array}}\right) }, \end{aligned}$$ which are new series representations for the Apery constant $$\zeta (3)$$ .

Good semigroups form a class of submonoids of Nd containing the value semigroups of curve singularities. In this article, we describe a partition of the complement of good semigroup ideals. As main...

Explicit algebraic area enumeration formulae are derived for various lattice walks generalizing the canonical square lattice walk, and in particular for the triangular lattice chiral walk recently introduced by the authors. A key element in the enumeration is the derivation of some remarkable identities involving trigonometric sums --which are also important building blocks of non trivial quantum models such as the Hofstadter model-- and their explicit rewriting in terms of multiple binomial sums. An intriguing connection is also made with number theory and some classes of Ap\'ery-like numbers, the cousins of the Ap\'ery numbers which play a central role in irrationality considerations for {\zeta}(2) and {\zeta}(3)., Comment: 31 pages, 4 figures

In this paper, we generalise a congruence which proved by V. J.-W. Guo and J. Zeng \cite{gz-jnt-2012} involving Ap\'{e}ry numbers, and we obtain a congruence involving Franel numbers which confirms a congruence conjecture of Z.-W. Sun \cite[Conjecture 57(ii)]{sun-njdx-2019}., Comment: 11pages

It has been conjectured that {\it all} graded Artinian Gorenstein algebras of codimension three have the weak Lefschetz property over a field of characteristic zero. In this paper, we study the weak Lefschetz property of associated graded algebras $A$ of the Ap\'ery set of $M$-pure symmetric numerical semigroups generated by four natural numbers. In 2010, Bryant proved that these algebras are graded Artinian Gorenstein algebras of codimension three. In a recent article, Guerrieri showed that if $A$ is not a complete intersection, then $A$ is of form $A=R/I$ with $R=K[x,y,z]$ and \begin{align*} I=(x^a, y^b-x^{b-\gamma} z^\gamma, z^c, x^{a-b+\gamma}y^{b-\beta}, y^{b-\beta}z^{c-\gamma}), \end{align*} where $ 1\leq \beta\leq b-1,\; \max\{1, b-a+1 \}\leq \gamma\leq \min \{b-1,c-1\}$ and $a\geq c\geq 2$. We prove that $A$ has the weak Lefschetz property in the following cases: (a) $ \max\{1,b-a+c-1\}\leq \beta\leq b-1$ and $\gamma\geq \lfloor\frac{\beta-a+b+c-2}{2}\rfloor$; (b) $ a\leq 2b-c$ and $| a-b| +c-1\leq \beta\leq b-1$; (c) one of $a,b,c$ is at most five., Comment: 20 pages. To appear in Linear Algebra and its Applications

We prove that the Apery constants for a certain class of Fano threefolds can be obtained as a special value of a higher normal function.

Given a generic $K3$-surface $Y_k$ of the Apery–Fermi pencil, we use the Kneser–Nishiyama technique to determine all its non isomorphic elliptic fibrations. These computations lead to determine those fibrations with 2-torsion sections T. We classify the fibrations such that the translation by T gives a Shioda–Inose structure. The other fibrations correspond to a $K3$-surface identified by its transcendental lattice. The same problem is solved for a singular member $Y_2$ of the family showing the differences with the generic case. In conclusion we put our results in the context of relations between 2-isogenies and isometries on the singular surfaces of the family.

We prove two conjectural congruences on the $(p-1)$ th Apéry number, which were recently proposed by Z.-H. Sun.

For n=0,1,2,… let Tn=∑k=0nnk22kn2 and Sn=∑k=0nnk2kk2n−2kn−k. Then {Tn} and {Sn} are Apery-like numbers. In this paper we obtain some congruences and pose several challenging conjectures for sums involving {Tn} or {Sn}.

We present a unified approach to obtain new series representations for various classical constants. Among others, we prove that $$\log (2) = \frac{{17}}{{24}} + \sum\limits_{k = 2}^\infty {{{( - 1)}^k}} \frac{{{k^2} + k - 1/2}}{{(k - 1)k(k + 1)(k + 2)}}({H_k} - {H_{{{\left[ {k/2} \right]}}})^2}$$ $$G = - \frac{1}{2} + 2\sum\limits_{k = 1}^\infty {{{( - 1)}^k}\frac{{k(4{k^2} - 5)}}{{(4{k^2} - 1)(4{k^2} - 9)}}{{(2{H_{2k}} - {H_k})}^2}} $$ , $$\zeta (3) = \frac{{149}}{{144}} + \frac{1}{8}\sum\limits_{k = 2}^\infty {\frac{{(2k + 1)({k^4} + 2{k^3} + 3{k^2} + 2k - 2)}}{{{{(k - 1)}^2}{k^2}{{(k + 1)}^2}(k + 2)}}{{(2H_k^{(2)} - H_{[k/2]}^{(2)})}^2}} $$ , where $${H_k} = \sum\nolimits_{j = 1}^k {1/j} $$ and $$H_k^{(2)} = {\sum\nolimits_{j = 1}^k {1/j} ^2}$$ denote the harmonic numbers and the generalized harmonic numbers of order 2, respectively, and G is the Catalan constant.

A numerical monoid is an additive submonoid of the non-negative integers. Given a numerical monoid S, consider the family of “shifted” monoids M n obtained by adding n to each generator of S. In this paper, we characterize the Apery set of M n in terms of the Apery set of the base monoid S when n is sufficiently large. We give a highly efficient algorithm for computing the Apery set of M n in this case, and prove that several numerical monoid invariants, such as the genus and Frobenius number, are eventually quasipolynomial as a function of n.

In this paper we confirm several conjectures of Z.-W. Sun on Hankel-type determinants for some combinatorial sequences including Franel numbers, Domb numbers and Ap\'ery numbers. For any nonnegative integer $n$, define \begin{gather*}f_n:=\sum_{k=0}^n\binom nk^3,\ D_n:=\sum_{k=0}^n\binom nk^2\binom{2k}k\binom{2(n-k)}{n-k}, b_n:=\sum_{k=0}^n\binom nk^2\binom{n+k}k,\ A_n:=\sum_{k=0}^n\binom nk^2\binom{n+k}k^2. \end{gather*} For $n=0,1,2,\ldots$, we show that $6^{-n}|f_{i+j}|_{0\leq i,j\leq n}$ and $12^{-n}|D_{i+j}|_{0\le i,j\le n}$ are positive odd integers, and $10^{-n}|b_{i+j}|_{0\leq i,j\leq n}$ and $24^{-n}|A_{i+j}|_{0\leq i,j\leq n}$ are always integers., Comment: 13 pages, final published version

In this paper, we generalize the work of Tuenter to give an identity which completely characterizes the complement of a numerical semigroup in terms of its Apery sets. Using this result, we compute the mth power Sylvester and alternating Sylvester sums for free numerical semigroups. Explicit formulas are given for small m.

The aim of the current document is to evaluate a quadruple integral involving the Chebyshev polynomial of the first kind Tn(x) and derive in terms of the Hurwitz-Lerch zeta function. Special cases are evaluated in terms of fundamental constants. The zero distribution of almost all Hurwitz-Lerch zeta functions is asymmetrical. All the results in this work are new.

A closed form expression for a triple integral not previously considered is derived, in terms of the Lerch function. Almost all Lerch functions have an asymmetrical zero-distribution. The kernel of the integral involves the product of the logarithmic, exponential, quotient radical, and polynomial functions. Special cases are derived in terms of fundamental constants, results are summarized in a table. All results in this work are new.