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Binomial sums via Bailey's cubic transformation

Wenchang Chu (2023)

Czechoslovak Mathematical Journal

By employing one of the cubic transformations (due to W. N. Bailey (1928)) for the 3 F 2 ( x ) -series, we examine a class of 3 F 2 ( 4 ) -series. Several closed formulae are established by means of differentiation, integration and contiguous relations. As applications, some remarkable binomial sums are explicitly evaluated, including one proposed recently as an open problem.

Binomials transformation formulae for scaled Fibonacci numbers

Edyta Hetmaniok, Bożena Piątek, Roman Wituła (2017)

Open Mathematics

The aim of the paper is to present the binomial transformation formulae of Fibonacci numbers scaled by complex multipliers. Many of these new and nontrivial relations follow from the fundamental properties of the so-called delta-Fibonacci numbers defined by Wituła and Słota. The paper contains some original relations connecting the values of delta-Fibonacci numbers with the respective values of Chebyshev polynomials of the first and second kind.

Boundedness of oriented walks generated by substitutions

F. M. Dekking, Z.-Y. Wen (1996)

Journal de théorie des nombres de Bordeaux

Let x = x 0 x 1 ⋯ be a fixed point of a substitution on the alphabet a , b , and let U a = - 1 - 1 0 1 and U b = 1 1 0 1 . We give a complete classification of the substitutions σ : a , b ☆ according to whether the sequence of matrices U x 0 U x 1 ⋯ U x n n = 0 ∞ is bounded or unbounded. This corresponds to the boundedness or unboundedness of the oriented walks generated by the substitutions.

Bounds for frequencies of residues of second-order recurrences modulo p r

Walter Carlip, Lawrence Somer (2007)

Mathematica Bohemica

The authors examine the frequency distribution of second-order recurrence sequences that are not p -regular, for an odd prime p , and apply their results to compute bounds for the frequencies of p -singular elements of p -regular second-order recurrences modulo powers of the prime p . The authors’ results have application to the p -stability of second-order recurrence sequences.

Bounds for the counting function of the Jordan-Pólya numbers

Jean-Marie De Koninck, Nicolas Doyon, A. Arthur Bonkli Razafindrasoanaivolala, William Verreault (2020)

Archivum Mathematicum

A positive integer n is said to be a Jordan-Pólya number if it can be written as a product of factorials. We obtain non-trivial lower and upper bounds for the number of Jordan-Pólya numbers not exceeding a given number x .

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