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We prove some congruences on sums involving fourth powers of central q -binomial coefficients. As a conclusion, we confirm the following supercongruence observed by Long [Pacific J. Math. 249 (2011), 405–418]:$$\\sum\\limits_{k = 0}^{((p^r-1)/(2))} {\\displaystyle{{4k + 1} \\over {{256}^k}}} \\left( \\matrix{2k \\cr k} \\right)^4\\equiv p^r\\quad \\left( {\\bmod p^{r + 3}} \\right),$$where p ⩾5 is a prime and r is a positive integer. Our method is similar to but a little different from the WZ method used by Zudilin to prove Ramanujan-type supercongruences.

By examining two hypergeometric series transformations, we establish several remarkable infinite series identities involving harmonic numbers and quintic central binomial coefficients, including five conjectured recently by Z.-W. Sun [‘Series with summands involving harmonic numbers’, Preprint, 2023, arXiv:2210.07238v7]. This is realised by ‘the coefficient extraction method’ implemented by Mathematica commands.

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 nth harmonic number and $B_n$ is the nth 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 $a, b$ .

We examine a useful hypergeometric transformation formula by means of the coefficient extraction method. A large class of “binomial/harmonic series” (of convergence ratio “1/4”) containing the cubic central binomial coefficients and harmonic numbers is systematically investigated. Numerous closed summation formulae are established, including a remarkable series about harmonic numbers of the third order.

We prove some finite sum identities involving reciprocals of the binomial and central binomial coefficients, as well as harmonic, Fibonacci and Lucas numbers, some of which recover previously known results, while the others are new.

By employing the coefficient extraction method, a class of binomial series involving harmonic numbers will be reviewed through three hypergeometric F12(y2)-series. Numerous closed-form generating functions for infinite series containing binomial coefficients and harmonic numbers will be established, including several conjectured ones.

Irreducible characters plays a crucial role in the representation theory of finite groups, namely the symmetric group 𝔊 n . In the current paper, we use the irreducible characters of 𝔊 n to define new weighted binomial coefficient sums. We use the Murnaghan-Nakayama rule to establish recurrence relations for these sums. As application, we employ the recurrences to derive explicit formulas for particular cases, for instance, Euler’s formula for the Stirling numbers of second kind is obtained in one of the particular cases. In the Appendix, we compute the initial values of the weighted sums.

By employing the generating function approach, 16 triple sums for circular binomial products of binomial coefficients are examined. Recurrence relations and generating functions are explicitly determined. These symmetric sums may find potential applications in the analysis of algorithms, symbolic calculus, and computations in theoretical physics.

Let q1+⋯+qn+m objects be arranged in n rows with q1,…,qn objects and one last row with m objects. The Janjić–Petković counting function denotes the number of (n+k)-insets, defined as subsets containing n+k objects such that at least one object is chosen from each of the first n rows, generalizing the binomial coefficient that is recovered for q1 = … = qn = 1, as then only the last row matters. Here, we discuss two explicit forms, combinatorial interpretations, recursion relations, an integral representation, generating functions, convolutions, special cases, and inverse pairs of summation formulas. Based on one of the generating functions, we show that the Janjić–Petković counting function, like the binomial coefficients that it generalizes, may be regarded as a Riordan array, leading to additional identities. As an application to a physical system, we calculate the heat capacity of a many-body system for which the configurations are constrained as described by the Janjić–Petković counting function, resulting in a modified Schottky anomaly.

By combining the generating function approach with the Lagrange expansion formula, we evaluate, in closed form, two multiple alternating sums of binomial coefficients, which can be regarded as alternating counterparts of the circular sum evaluation discovered by Carlitz [‘The characteristic polynomial of a certain matrix of binomial coefficients’, Fibonacci Quart. 3(2) (1965), 81–89].