Products and Sums of Powers of Binomial Coefficients mod p and Solutions of Certain Quaternary Diophantine Systems

1984 ◽  
Vol 43 (168) ◽  
pp. 603
Author(s):  
Richard H. Hudson
Keyword(s):  
Binomial Coefficients ◽  
Sums Of Powers ◽  
Mod P

On the rate of p-adic convergence of sums of powers of binomial coefficients

2017 ◽  
Vol 13 (08) ◽  
pp. 2075-2091 ◽  
Author(s):  
Tamás Lengyel
Keyword(s):  
Point Of View ◽  
Binomial Coefficients ◽  
Arithmetic Properties ◽  
Sums Of Powers ◽  
Divisibility Properties

Let [Formula: see text] be an integer and [Formula: see text] be an odd prime. We study sums and lacunary sums of [Formula: see text]th powers of binomial coefficients from the point of view of arithmetic properties. We develop new congruences and prove the [Formula: see text]-adic convergence of some subsequences and that in every step we gain at least one or three more [Formula: see text]-adic digits of the limit if [Formula: see text] or [Formula: see text], respectively. These gains are exact under some explicitly given conditions. The main tools are congruential and divisibility properties of the binomial coefficients and multiple and alternating harmonic sums.

P-Adic Binomial Coefficients Mod P

1985 ◽  
Vol 92 (8) ◽  
pp. 576-578 ◽  
Author(s):  
Roger C. Alperin
Keyword(s):  
Binomial Coefficients ◽  
Mod P

scholarly journals On recurrences for sums of powers of binomial coefficients

2008 ◽  
Vol 128 (10) ◽  
pp. 2784-2794 ◽  
Author(s):  
Yuan Jin ◽  
Zhi-Juan Lu ◽  
Asmus L. Schmidt
Keyword(s):  
Binomial Coefficients ◽  
Sums Of Powers

scholarly journals Generalized Stirling Numbers and Hyper-Sums of Powers of Binomials Coefficients

10.37236/3787 ◽  
2014 ◽  
Vol 21 (1) ◽  
Author(s):  
Claudio de J. Pita-Ruiz V.
Keyword(s):  
Stirling Numbers ◽  
Binomial Coefficients ◽  
Partial Sums ◽  
Linear Combinations ◽  
Normal Ordering ◽  
Sums Of Powers ◽  
Generalized Stirling Numbers ◽  
Natural Way

We work with a generalization of Stirling numbers of the second kind related to the boson normal ordering problem (P. Blasiak et al.). We show that these numbers appear as part of the coefficients of expressions in which certain sequences of products of binomials, together with their partial sums, are written as linear combinations of some other binomials. We show that the number arrays formed by these coefficients can be seen as natural generalizations of Pascal and Lucas triangles, since many of the known properties on rows, columns, falling diagonals and rising diagonals in Pascal and Lucas triangles, are also valid (some natural generalizations of them) in the arrays considered in this work. We also show that certain closed formulas for hyper-sums of powers of binomial coefficients appear in a natural way in these arrays.

scholarly journals Supercongruences involving Apéry-like numbers and binomial coefficients

2021 ◽  
Vol 7 (2) ◽  
pp. 2729-2781
Author(s):  
Zhi-Hong Sun ◽  
Keyword(s):  
Binomial Coefficients ◽  
Mod P

<abstract><p>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.</p></abstract>

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