Published Papers and Preprints

You can see my Google Scholar profile HERE, and my Research Gate profile HERE.
My ORCID ID is 0000-0002-5619-3513. My Erdos Number is 3.

• Arithmetic properties of MacMahon-type sums of divisors (with James A. Sellers)
  The Ramanujan Journal, 67 (2025), article 37, 1-17 .
  In this paper, we prove several new infinite families of Ramanujan--like congruences satisfied by the coefficients of the generating function U_t(a,q) which is an extension of MacMahon's generalized sum-of-divisors function. As a by-product, we also show that, for all nonnegative integers n, B_3(15n+7)=0 (mod5) where B_3(n) is the number of almost 3-regular overpartitions of n.

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• More q-analogs of congruences involving Catalan numbers)
  Integral Transforms and Special Functions (2025), 1-9 .
  We investigate q-analogs of congruences \sum_{k=0}^{p^r-1}C_k/3^k and \sum_{k=1}^{p-1}C_k/(k3^k) modulo a prime p. The main tool for our proofs is a q-identity established by G. E. Andrews.

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• Further study on MacMahon-type sums of divisors (with Tewodros Amdeberhan and George E. Andrews)
  Research in Number Theory, 11 (2025), article 19, 1-17.
  This paper is devoted to the study of U_t(a,q) when a is one of 0,1,-1,2,-2. The idea builds on our previous treatment of the case a=-2. It is shown that all these functions lie in the ring of quasi-modular forms. Among the more surprising findings is U_2(1,q)=\sum_{n\geq1} \frac{q^{3n}}{(1-q^{3n})^2}.

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• Congruences for sums of MacMahon's q-Catalan polynomials (Tewodros Amdeberhan)
  Bulletin of the Australian Mathematical Society (First published online)
  One variant of the q-Catalan polynomials is defined in terms of Gaussian polynomials by C_k(q)=\qbin{2k}{k}-q\qbin{2k}{k+1}. Liu studied congruences of the form sum_{k=0}^{n-1} q^k C_k(q) modulo the cyclotomic polynomial Phi_n(q)^2, provided that n\equiv\pm 1\pmod3. Apparently the case n\equiv 0\pmod3 has been missing from the literature. It is our primary purpose to fill this gap by the current work. In addition, we discuss certain fascinating link to Dirichlet character sum identities.

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• Extensions of MacMahon's sums of divisors (with Tewodros Amdeberhan and George E. Andrews)
  Research in the Mathematical Sciences, 11 (2024), article 8, 1-23.
  In 1920, P. A. MacMahon generalized the (classical) notion of divisor sums by relating it to the theory of partitions of integers. In this paper, we extend the idea of MacMahon. In doing so we reveal a wealth of divisibility theorems and unexpected combinatorial identities. Our initial approach is quite different from MacMahon and involves rational function approximation to MacMahon-type generating functions.

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• Congruences for partial sums of the generating series for binomial(3k,k) (with Sandro Mattarei)
  International Journal of Number Theory, 21 (2025), 237-256
  We produce congruences modulo a prime p > 3 for sums Σ binomial(3k,k)x^k over ranges 0 ≤ k < q and 0 ≤ k < q/3, where q is a power of p. Here x equals either c^2/(1-c)^3, or 4s^2/(27(s^2-1)), where c and s are indeterminates. In the former case we deal more generally with shifted binomial coefficients binomial(3k+e,k). Our method derives such congruences directly from closed forms for the corresponding series.

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• More Congruences for Central Binomial Sums with Fibonacci and Lucas Numbers
  Journal of Integer Sequences, 26 (2023), 23.8.3.
  We mainly determine sum_{k=1}^{p-1}\binom{2k}{k}h_kx^k modulo a prime p with h_k=sum_{j=1}^{k}1/(2j-1). We also provide some applications of this polynomial congruence for some special values of x which involve the Fibonacci and Lucas numbers.

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• A local-global theorem for p-adic supercongruences (with Hao Pan and Chen Wang)
  Journal fur die reine und angewandte Mathematik, 790 (2022), 53-83.
  We prove that if a sufficiently regular n-variable function is zero modulo p^r over some suitable collection of r hyperplanes, then it is zero modulo p^r over the whole (Z_p)^n. We provide various applications of this general criterion by establishing several p-adic analogues of hypergeometric identities.

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• On 3-2-1 values of finite multiple harmonic q-series at roots of unity (with Khodabakhsh Hessami Pilehrood and Tatiana Hessami Pilehrood)
  Journal of the Mathematical Society of Japan, 74 (2022), No. 3, 753-758.
  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.

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• Three pairs of congruences concerning sums of central binomial coefficients (with Guo-Shuai Mao)
  International Journal of Number Theory, 17 (2021), No. 10, 2301-2314.
  We prove six congruences modulo p^2 concerning sums of central binomial coefficients where the upper limit is strictly between (p-1)/2 and p-1.

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• Two supercongruences related to multiple harmonic sums
  Bulletin of the Australian Mathematical Society, 103 (2021), 379-389.
  Let p be a prime and let x be a p-adic integer. We provide two supercongruences for truncated series involving Pochhammer symbols and multiple (non-strict) harmonic sums.

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• A bivariate generating function for zeta values and related supercongruences
  Journal of Difference Equations and Applications, 26 (2020), 1526-1537.
  By using the Wilf-Zeilberger method, we prove a novel finite combinatorial identity related to a bivariate generating function for zeta(2+r+2s) (an extension of a Bailey-Borwein-Bradley Apery-like formula for even zeta values). Such identity is then applied to show several supercongruences.

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• A supercongruence involving cubes of Catalan numbers
  INTEGERS, 20 (2020), A44.
  We mainly show a supercongruence for a truncated series with cubes of Catalan numbers which extends a result by Zhi-Wei Sun.

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• A congruence for a double harmonic sum (with Tewodros Amdeberhan)
  Scientia, Ser. A, Math. Sci. (N.S.), 29 (2019), 37-44.
  In this short note, our primary purpose is to prove a congruence modulo p for a double harmonic sum. Along the way, a number of auxiliary results of independent interest are found.

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• Supercongruences related to 3F2(1) involving harmonic numbers
  International Journal of Number Theory, 14 (2018), 1093-1109.
  We show various supercongruences for truncated series which involve central binomial coefficients and harmonic numbers. The corresponding infinite series are also evaluated.

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• From generating series to polynomial congruences (with Sandro Mattarei)
  Journal of Number Theory, 182 (2018), 179-205.
  Consider an ordinary generating function sum_{k=0}^{infty}c_kx^k, of an integer sequence of some combinatorial relevance, and assume that it admits a closed form C(x). Various instances are known where the corresponding truncated sum sum_{k=0}^{q-1}c_kx^k, with q a power of a prime p, also admits a closed form representation when viewed modulo p. Such a representation for the truncated sum modulo p frequently bears a resemblance with the shape of C(x), despite being typically proved through independent arguments. We develop a method which allows one to directly infer the closed-form representation of the truncated sum from the closed form of the series for a significant class of series involving central binomial coefficients.

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• Revitalized automatic proofs: demonstrations (with Tewodros Amdeberhan, David Callan, and Hideyuki Ohtsuka)
  INTEGERS, 17 (2017), A16.
  We consider three problems from the recent issues of the American Mathematical Monthly involving different versions of Catalan triangle. Our main results offer generalizations of these identities and demonstrate automated proofs with additional twists, and on occasion we furnish a combinatorial proof.

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• Two Triple binomial sum supercongruences (with Tewodros Amdeberhan)
  Journal of Number Theory, 175 (2017), 140-157.
  In a recent article, Apagodu and Zeilberger discuss some applications of an algorithm for finding and proving congruence identities (modulo primes) of indefinite sums of many combinatorial sequence. At the end, they propose some supercongruences as conjectures. Here we prove one of them, including a new companion enumerating abelian squares, and we leave some remarks for the others. This joint work extends this previous draft.

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• Multiple harmonic sums and multiple harmonic star sums are (nearly) never integers (with Khodabakhsh Hessami Pilehrood and Tatiana Hessami Pilehrood)
  INTEGERS, 17 (2017), A10.
  A well known result of elementary number theory is that even though the partial sum H_n of the harmonic series increases to infinity, it is never an integer for n>1. Apparently the first published proof goes back to Leopold Theisinger in 1915, and, since then, it has been proposed as a challenging problem in several textbooks. In 1946, Erdos and Niven proved a stronger statement: there is only a finite number of integers n for which there is a positive integer r<=n such that the r-th elementary symmetric function of 1,1/2,...,1/n is an integer. In 2012, Chen and Tang refined this result and succeeded to show that the above sum is not an integer with the only two exceptions: either n=r=1 or n=3 and r=2. In this paper, we consider the integrality problem for sums which are not necessarily symmetric with respect to their variables: the multiple harmonic and multiple harmonic star sums.

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• Some q-congruences for homogeneous and quasi-homogeneous multiple q-harmonic sums (with Khodabakhsh Hessami Pilehrood and Tatiana Hessami Pilehrood)
  Ramanujan Journal, 43 (2017), 113-139.
  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.

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• Restricted linear congruences (with Khodakhast Bibak, Bruce M. Kapron, Venkatesh Srinivasan, and Laszlo Toth)
  Journal of Number Theory, 171 (2017), 128-144.
  In this paper, using properties of Ramanujan sums and of the finite Fourier transform of arithmetic functions, we give an explicit formula for the number of solutions of the linear congruence a_1*x_1+...+a_k*x_k=b (mod n), with (x_i,n)=t_i for 1<=i<=k. Some special cases of this problem have been studied in many papers, and have found very interesting applications in number theory, combinatorics, and cryptography, among other areas. We also propose an authenticated encryption scheme, and using our explicit formula, analyze the integrity of this scheme.

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• Large Peg-Army Maneuvers (with Luciano Guala', Stefano Leucci, and Emanuele Natale)
  8th International Conference on Fun with Algorithms (FUN 2016), Volume 49, FUN 2016, June 8-10, 2016 - La Maddalena, Italy.
  Despite its long history, the classical game of peg solitaire continues to attract the attention of the scientific community. In this paper, we consider two problems with an algorithmic flavour which are related with this game, namely Solitaire-Reachability and Solitaire-Army. In the first one, we show that deciding whether there is a sequence of jumps which allows a given initial configuration of pegs to reach a target position is NP-complete. Regarding Solitaire-Army, the aim is to successfully deploy an army of pegs in a given region of the board in order to reach a target position. By solving an auxiliary problem with relaxed constraints, we are able to answer some open questions raised by Csakany and Juhasz (Mathematics Magazine, 2000). See the gif-animation given HERE.

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• Supercongruences for the Almkvist-Zudilin numbers (with Tewodros Amdeberhan)
  Acta Arithmetica, 173 (2016), 255-268.
  Given a prime number p, the study of divisibility properties of a sequence c(n) has two contending approaches: p-adic valuations and superconcongruences. The former searches for the highest power of p dividing c(n), for each n; while the latter (essentially) focuses on the maximal powers r and t such that c(p^rn) is congruent to c(p^{r-1}n) modulo p^t. This is called supercongruence. In this paper, we prove a conjecture on supercongruences for sequences that have come to be known as the Almkvist-Zudilin numbers. Some other (naturally) related family of sequences will be considered in a similar vain.

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• Some congruences for central binomial sums involving Fibonacci and Lucas numbers
  Journal of Integer Sequences, 19 (2016), 16.5.4.
  We present several polynomial congruences about sums with central binomial coefficients and harmonic numbers. In the final section we collectsome new congruences involving Fibonacci and Lucas numbers.

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• New multiple harmonic sum identities (with Helmut Prodinger)
  Electronic Journal of Combinatorics, 21 (2014), P2.43.
  We consider a special class of binomial sums involving harmonic numbers and we prove three identities by using the elementary method of the partial fraction decomposition. Some applications to infinite series and congruences are given.

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• New properties of multiple harmonic sums modulo p and p-analogues of Leshchiner's series (with Khodabakhsh Hessami Pilehrood and Tatiana Hessami Pilehrood).
  Transactions of the American Mathematical Society, 366 (2014), 3131-3159.
  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 \zeta^{*}(\{2\}^a,3,\{2\}^b) based on a finite identity for partial sums of the zeta-star series.

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• Determinants of grids, tori, cylinders and Mobius ladders (with Khodakhast Bibak)
  Discrete Mathematics, 313 (2013), 1436-1440.
  Recently, Bien [A. Bien, The problem of singularity for planar grids, Discrete Math. 311 (2011), 921--931] obtained a recursive formula for the determinant of a grid. Also, recently, Pragel [D. Pragel, Determinants of box products of paths, Discrete Math. 312 (2012), 1844--1847], independently, obtained an explicit formula for this determinant. In this paper, we give a short proof for this problem. Furthermore, applying the same technique, we get explicit formulas for the determinant of a torus, a cylinder, and a Mobius ladder.

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• Some q-analogs of congruences for central binomial sums
  Colloquium Mathematicum, 133 (2013), 133-143.
  We establish q-analogs for four congruences involving central binomial coefficients. The q-identities necessary for this purpose are shown via the q-WZ method.

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• Congruences for central binomial sums and finite polylogarithms (with Sandro Mattarei)
  Journal of Number Theory, 133 (2013), 131-157.
  We prove congruences, modulo a power of a prime p, for certain finite sums involving central binomial coefficients \binom{2k}{k} in the denominators. Specifically, we evaluate the sums p\sum_{k=1}^{p-1} {t^k k^{-d}\binom{2k}{k}^{-1}}\pmod{p^3} and p\sum_{k=1}^{p-1} {t^k H_{k-1}(2)\,k^{-d}\binom{2k}{k}^{-1}}\pmod{p}, for t\in\{1,-1,2,3,4,-1/2\} and d=0,1,2.

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• Congruences concerning Jacobi polynomials and Apery-like formulae (with Khodabakhsh Hessami Pilehrood and Tatiana Hessami Pilehrood)
  International Journal of Number Theory, 8 (2012), 1789-1811.
  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}. Part of this paper follows from arXiv:1110.4013.

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• Supercongruences for a truncated hypergeometric series
  INTEGERS, 12 (2012), A45.
  The purpose of this note is to obtain some congruences modulo a power of a prime p involving the truncated hypergeometric series \sum_{k=1}^{p-1} {(x)_k(1-x)_k\over (1)_k^2}\cdot{1\over k^a} for a=1 and a=2. In the last section, the special case x=1/2 is considered.

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• q-Analogs of some congruences involving Catalan numbers
  Advances in Applied Mathematics, 48 (2012), 603-614.
  We provide some variations on the Greene-Krammer's identity which involve q-Catalan numbers. Our method reveals a curious analogy between these new identities and some congruences modulo a prime.

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• On some new congruences for binomial coefficients (with Zhi-Wei Sun)
  International Journal of Number Theory, 7 (2011), 645-662.
  In this paper we establish some new congruences involving Catalan numbers as well as central binomial coefficients. Let p>3 be a prime. We show that sum_{k=0}^{p-1}C_{pn+k}/C_n=1-3(n+1)((p-1)/3) (mod p^2) for every n=0,1,2,..., where C_m is the Catalan number binom(2m,m)/(m+1), and (*/3) is the Legendre symbol. We also determine sum_{k=0}^{p^a-1}binom(2k,k+d) and sum_{k=0}^{p^a-1}k*binom(2k,k+d) modulo p^2 for all a=1,2,3,... and d=0,1,...,p.

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• Constraining spacetime torsion with the Moon and Mercury (with Riccardo March, Giovanni Bellettini and Simone Dell'Agnello)
  Phys. Rev. D 83, 104008 (2011).
  We report a search for new gravitational physics phenomena based on Einstein-Cartan theory of General Relativity including spacetime torsion. Starting from the parametrized torsion framework of Mao, Tegmark, Guth and Cabi, we analyze the motion of test bodies in the presence of torsion, and in particular we compute the corrections to the perihelion advance and to the orbital geodetic precession of a satellite.

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• Constraining spacetime torsion with LAGEOS (with Riccardo March, Giovanni Bellettini and Simone Dell'Agnello)
  General Relativity and Gravitation, 43 (2011), Number 11, 3099-3126.
  We compute the corrections to the orbital Lense-Thirring effect (or frame-dragging) in the presence of spacetime torsion. We derive the equations of motion of a test body in the gravitational field of a rotating axisymmetric massive body, using the parametrized framework of Mao, Tegmark, Guth and Cabi. We calculate the secular variations of the longitudes of the node and of the pericenter.

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