Results 21 to 30 of about 100,720 (284)

PROOF OF TWO CONJECTURES ON SUPERCONGRUENCES INVOLVING CENTRAL BINOMIAL COEFFICIENTS

Bulletin of the Australian Mathematical Society, 2020
In this note we use some $q$-congruences proved by the method of ‘creative microscoping’ to prove two conjectures on supercongruences involving central binomial coefficients. In particular, we confirm the $m=5$ case of Conjecture 1.1 of Guo [‘Some generalizations of a supercongruence of Van Hamme’, Integral Transforms Spec. Funct.28 (2017), 888–899].
Chengyang Gu, Victor J. W. Guo
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Two permutation classes enumerated by the central binomial coefficients [PDF]

, 2013
We define a map between the set of permutations that avoid either the four patterns $3214,3241,4213,4231$ or $3124,3142,4123,4132$, and the set of Dyck prefixes. This map, when restricted to either of the two classes, turns out to be a bijection that allows us to determine some notable features of these permutations, such as the distribution of the ...
Marilena Barnabei, Flavio Bonetti, Matteo Silimbani   +2 more
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Overlapping genes connect rheumatoid arthritis and head and neck cancer: coincidence or shared immune pathophysiology? [PDF]

Front Med (Lausanne)
IntroductionDespite advances in understanding the pathophysiology of rheumatoid arthritis (RA) and head and neck cancer (HNC) individually, their shared genetic and molecular mechanisms remain poorly defined.MethodsThis study aimed to explore gene-level ...
Wang R, Li H, Yang Y, Lian M.
europepmc   +2 more sources

Three combinatorial sums involving central binomial coefficients [PDF]

We study three classes of combinatorial sums involving central binomial coefficients and harmonic numbers, odd harmonic numbers, and even indexed harmonic numbers, respectively. In each case we use summation by parts to derive recursive expressions for these sums.
Kunle Adegoke, Robert Frontczak
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Infinite series containing quotients of central binomial coefficients [PDF]

Notes on Number Theory and Discrete Mathematics
By making use of the Wallis' integral formulae and integration by parts, two classes of infinite series are evaluated, in closed form, in terms of π and Riemann zeta function.
Zhiling Fan
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Further Classes of Series Involving Central Binomial Coefficients [PDF]

Departing from a class of infinite series with central binomial coefficients in the numerator and depending on a positive integer parameter, we first extend known identities to all complex parameters. Then we use various methods, including exponential Bell polynomials and integral representations, to further extend these results.
Karl Dilcher, Christophe Vignat
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The Series of Reciprocals of Non-central Binomial Coefficients

American Journal of Computational Mathematics, 2013
Utilizing Gamma-Beta function, we can build one series involving reciprocal of non-central binomial coefficients, then We can structure several new series of reciprocals of non-central binomial coefficients by item splitting, these new created denominator of series contain 1 to 4 odd factors of binomial coefficients.
Laiping Zhang, JI Wan-hui
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Binomial Sum Relations Involving Fibonacci and Lucas Numbers

AppliedMath, 2023
In this paper, we provide a first systematic treatment of binomial sum relations involving (generalized) Fibonacci and Lucas numbers. The paper introduces various classes of relations involving (generalized) Fibonacci and Lucas numbers and different ...
Kunle Adegoke, Robert Frontczak, Taras Goy   +2 more
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Infinite series about harmonic numbers inspired by Ramanujan–like formulae

Electronic Research Archive, 2023
By employing the coefficient extraction method from hypergeometric series, we shall establish numerous closed form evaluations for infinite series containing central binomial coefficients and harmonic numbers, including several conjectured ones made by Z.
Chunli Li, Wenchang Chu
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Series of Convergence Rate −1/4 Containing Harmonic Numbers

Axioms, 2023
Two general transformations for hypergeometric series are examined by means of the coefficient extraction method. Several interesting closed formulae are shown for infinite series containing harmonic numbers and binomial/multinomial coefficients.
Chunli Li, Wenchang Chu
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