arXiv subjects
Publications and source records attributed to Khodabakhsh Hessami Pilehrood.
We mainly answer two open questions about finite multiple harmonic $q$-series on 3-2-1 indices at roots of unity, posed recently by H. Bachmann, Y. Takeyama, and K. Tasaka. Two conjectures regarding cyclic sums which generalize the given results are also provided.
We study generating functions for multiple zeta star values in general form. These generating functions provide a connection between multiple zeta star values and multiple Euler sums, which allows us to express each multiple zeta star value in terms of multiple alternating Euler sums, and specifically, reduce the length of blocks of twos in the resulting sums.
In 2008, Muneta found explicit evaluation of the multiple zeta star value $ζ^\star(\{3, 1\}^d)$, and in 2013, Yamamoto proved a sum formula for multiple zeta star values on 3-2-1 indices. In this paper, we provide another way of deriving the formulas mentioned above. It is based on our previous work on generating functions for multiple zeta star values and also on constructions of generating functions for restricted sums of alternating Euler sums. As a result, the formulas obtained are simpler and computationally more effective than the known ones. Moreover, we give explicit evaluations of $ζ^\star(\{\{2\}^m, 3, \{2\}^m, 1\bigr\}^d)$ and $ζ^\star(\{\{2\}^m, 3, \{2\}^m, 1\}^d, \{2\}^{m+1})$ in two ways. The first is based on computation of product of generating functions, while the second uses properties of Bell polynomials.
It is well known that the harmonic sum $H_n(1)=\sum_{k=1}^n\frac{1}{k}$ is never an integer for $n>1$. In 1946, Erdős and Niven proved that the nested multiple harmonic sum $H_n(\{1\}^r)=\sum_{1\le k_1<\dots<k_r\le n}\frac{1}{k_1\cdots k_r}$ can take integer values only for a finite number of positive integers $n$. In 2012, Chen and Tang refined this result by showing that $H_n(\{1\}^r)$ is an integer only for $(n,r)=(1,1)$ and $(n,r)=(3,2)$. In this paper, we consider the integrality problem for arbitrary multiple harmonic and multiple harmonic star sums and show that none of these sums is an integer with some natural exceptions like those mentioned above.
In recent years, there has been intensive research on the ${\mathbb Q}$-linear relations between multiple zeta (star) values. In this paper, we prove many families of identities involving the $q$-analog of these values, from which we can always recover the corresponding classical identities by taking $q\to 1$. The main result of the paper is the duality relations between multiple zeta star values and Euler sums and their $q$-analogs, which are generalizations of the Two-one formula and some multiple harmonic sum identities and their $q$-analogs proved by the authors recently. Such duality relations lead to a proof of the conjecture by Ihara et al. that the Hoffman $\star$-elements $ζ^{\star}(s_1,\dots,s_r)$ with $s_i\in\{2,3\}$ span the vector space generated by multiple zeta values over ${\mathbb Q}$.
In this paper, we study congruences on sums of products of binomial coefficients that can be proved by using properties of the Jacobi polynomials. We give special attention to polynomial congruences containing Catalan numbers, second-order Catalan numbers, the sequence (\seqnum{A176898}) $S_n=\frac{{6n\choose 3n}{3n\choose 2n}}{2{2n\choose n}(2n+1)},$ and the binomial coefficients ${3n\choose n}$ and ${4n\choose 2n}$. As an application, we address several conjectures of Z.\ W.\ Sun on congruences of sums involving $S_n$ and we prove a cubic residuacity criterion in terms of sums of the binomial coefficients ${3n\choose n}$ conjectured by Z.\ H.\ Sun.
We show some new Wolstenholme type $q$-congruences for some classes of multiple $q$-harmonic sums of arbitrary depth with strings of indices composed of ones, twos and threes. Most of these results are $q$-extensions of the corresponding congruences for ordinary multiple harmonic sums obtained by the authors in a previous paper. Finally, we pose a conjecture concerning two kinds of cyclic sums of multiple $q$-harmonic sums.
Recently, A. I. Aptekarev and his collaborators found a sequence of rational approximations to Euler's constant $γ$ defined by a third-order homogeneous linear recurrence. In this paper, we give a new interpretation of Aptekarev's approximations in terms of Meijer $G$-functions and hypergeometric-type series. This approach allows us to describe a very general construction giving linear forms in 1 and $γ$ with rational coefficients. Using this construction we find new rational approximations to $γ$ generated by a second-order inhomogeneous linear recurrence with polynomial coefficients. This leads to a continued fraction (though not a simple continued fraction) for Euler's constant. It seems to be the first non-trivial continued fraction expansion convergent to Euler's constant sub-exponentially, the elements of which can be expressed as a general pattern. It is interesting to note that the same homogeneous recurrence generates a continued fraction for the Euler-Gompertz constant found by Stieltjes in 1895.
Recently, the present authors jointly with Tauraso found a family of binomial identities for multiple harmonic sums (MHS) on strings $(\{2\}^a,c,\{2\}^b)$ that appeared to be useful for proving new congruences for MHS as well as new relations for multiple zeta values. Very recently, Zhao generalized this set of MHS identities to strings with repetitions of the above patterns and, as an application, proved the two-one formula for multiple zeta star values conjectured by Ohno and Zudilin. In this paper, we extend our approach to $q$-binomial identities and prove $q$-analogues of two-one formulas for multiple zeta star values.
In this paper we present some new identities of hypergeometric type for multiple harmonic sums whose indices are the sequences $(\{1\}^a,c,\{1\}^b),$ $(\{2\}^a,c,\{2\}^b)$ and prove a number of congruences for these sums modulo a prime $p.$ The congruences obtained allow us to find nice $p$-analogues of Leshchiner's series for zeta values and to refine a result due to M. Hoffman and J. Zhao about the set of generators of the multiple harmonic sums of weight 7 and 9 modulo $p$. Moreover, we are also able to provide a new proof of Zagier's formula for $ζ^{*}(\{2\}^a,3,\{2\}^b)$ based on a finite identity for partial sums of the zeta-star series.
Let $p>5$ be a prime. We prove congruences modulo $p^{3-d}$ for sums of the general form $\sum_{k=0}^{(p-3)/2}\binom{2k}{k}t^k/(2k+1)^{d+1}$ and $\sum_{k=1}^{(p-1)/2}\binom{2k}{k}t^k/k^d$ with $d=0,1$. We also consider the special case $t=(-1)^{d}/16$ of the former sum, where the congruences hold modulo $p^{5-d}$.
Recently, R. Tauraso established finite $p$-analogues of famous Apéry series for $ζ(2)$ and $ζ(3).$ In this paper, we present several congruences for finite central binomial sums arising from the truncation of Apéry-type series for $ζ(4)$ and $ζ(5).$ We also prove a $p$-analogue of Zeilberger's series for $ζ(2)$ confirming a conjecture of Z. W. Sun.
In this paper, we prove a new identity for values of the Hurwitz zeta function which contains as particular cases Koecher's identity for odd zeta values, the Bailey-Borwein-Bradley identity for even zeta values and many other interesting formulas related to values of the Hurwitz zeta function. We also get an extension of the bivariate identity of Cohen to values of the Hurwitz zeta function. The main tool we use here is a construction of new Markov-WZ pairs. As application of our results, we prove several conjectures on supercongruences proposed by J. Guillera, W. Zudilin, and Z.-W. Sun.
In this paper, we continue to study properties of rational approximations to Euler's constant and values of the Gamma function defined by linear recurrences, which were found recently by A. I. Aptekarev and T. Rivoal. Using multiple Jacobi-Laguerre orthogonal polynomials we present rational approximations to the quotient of values of the Gamma function at rational points. As a limit case of our result, we obtain new explicit formulas for numerators and denominators of the Aptekarev approximants to Euler's constant.
In this paper, we present new explicit simultaneous rational approximations converging sub-exponentially to the values of Bell polynomials at the points of the form $(γ, 1! (2a+1)ζ(2), 2!ζ(3),..., (m-1)!(a+1+(-1)^ma)ζ(m)),$ $m=1,2,...,a,$ $a\in{\mathbb N}.$
In 2007, A.I.Aptekarev and his collaborators discovered a sequence of rational approximations to Euler's constant $γ$ defined by a linear recurrence. In this paper, we generalize this result and present an explicit construction of rational approximations for the numbers $\ln(b)-ψ(a+1),$ $a, b\in {\mathbb Q},$ $b>0, a>-1,$ where $ψ$ defines the logarithmic derivative of the Euler gamma function. We prove exact formulas for denominators and numerators of the approximations in terms of hypergeometric sums. As a consequence, we get rational approximations for the numbers $π/2\pmγ.$ We compare the results obtained with those of T. Rivoal for the numbers $γ+\ln(b)$ and prove denominators conjectures proposed by Rivoal for denominators of rational approximations for $γ+\ln(b)$ and common denominators of simultaneous approximations for the numbers $γ$ and $ζ(2)-γ^2.$
We establish a $q$-analogue of the Bailey-Borwein-Bradley identity generating accelerated series for even zeta values and prove $q$-analogues of Markov's and Amdeberhan's series for $ζ(3)$ using the $q$-Markov-WZ method.
We generalize well-known Catalan-type integrals for Euler's constant to values of the generalized-Euler-constant function and its derivatives. Using generating functions appeared in these integral representations we give new Vacca and Ramanujan-type series for values of the generalized-Euler-constant function and Addison-type series for values of the generalized-Euler-constant function and its derivative. As a consequence, we get base $B$ rational series for $\log\frac{4}π,$ $\frac{G}π$ (where $G$ is Catalan's constant), $\frac{ζ'(2)}{π^2}$ and also for logarithms of Somos's and Glaisher-Kinkelin's constants.