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Approximation properties of bivariate complex q -Bernstein polynomials in the case q > 1

Nazim I. Mahmudov (2012)

Czechoslovak Mathematical Journal

In the paper, we discuss convergence properties and Voronovskaja type theorem for bivariate q -Bernstein polynomials for a function analytic in the polydisc D R 1 × D R 2 = { z ∈ C : | z | < R 1 } × { z ∈ C : | z | < R 1 } for arbitrary fixed q > 1 . We give quantitative Voronovskaja type estimates for the bivariate q -Bernstein polynomials for q > 1 . In the univariate case the similar results were obtained by S. Ostrovska: q -Bernstein polynomials and their iterates. J. Approximation Theory 123 (2003), 232–255. and S. G. Gal: Approximation by Complex Bernstein and Convolution...

Arithmetic of linear forms involving odd zeta values

Wadim Zudilin (2004)

Journal de Théorie des Nombres de Bordeaux

A general hypergeometric construction of linear forms in (odd) zeta values is presented. The construction allows to recover the records of Rhin and Viola for the irrationality measures of ζ ( 2 ) and ζ ( 3 ) , as well as to explain Rivoal’s recent result on infiniteness of irrational numbers in the set of odd zeta values, and to prove that at least one of the four numbers ζ ( 5 ) , ζ ( 7 ) , ζ ( 9 ) , and ζ ( 11 ) is irrational.

Deformed Heisenberg algebra with reflection and d -orthogonal polynomials

Fethi Bouzeffour, Hanen Ben Mansour, Ali Zaghouani (2017)

Czechoslovak Mathematical Journal

This paper is devoted to the study of matrix elements of irreducible representations of the enveloping deformed Heisenberg algebra with reflection, motivated by recurrence relations satisfied by hypergeometric functions. It is shown that the matrix elements of a suitable operator given as a product of exponential functions are expressed in terms of d -orthogonal polynomials, which are reduced to the orthogonal Meixner polynomials when d = 1 . The underlying algebraic framework allowed a systematic derivation...

Développements asymptotiques q -Gevrey et séries G q -sommables

Changgui Zhang (1999)

Annales de l'institut Fourier

Nous donnons une version q -analogue de l’asymptotique Gevrey et de la sommabilité de Borel, dues respectivement à G. Watson et E. Borel et systématiquement développées depuis une quinzaine d’années par J.-P. Ramis, Y. Sibuya, etc. Le but de ces auteurs était l’étude des équations différentielles dans le champ complexe. De même notre but est l’étude des équations aux q -différences dans le champ complexe, dans la ligne de G.D. Birkhoff et W.J. Trjitzinsky.Plus précisément, nous introduisons une nouvelle...

Divisors, partitions and some new q-series identities

Alexander E. Patkowski (2009)

Colloquium Mathematicae

We obtain new q-series identities that have interesting interpretations in terms of divisors and partitions. We present a proof of a theorem of Z. B. Wang, R. Fokkink, and W. Fokkink, which follows as a corollary to our main q-series identity, and offer a similar result.

Exotic Bailey-Slater spt-functions III: Bailey pairs from groups B, F, G, and J

Chris Jennings-Shaffer (2016)

Acta Arithmetica

We continue to investigate spt-type functions that arise from Bailey pairs. In this third paper on the subject, we proceed to introduce additional spt-type functions. We prove simple Ramanujan type congruences for these functions which can be explained by an spt-crank-type function. The spt-crank-type functions are actually defined first, with the spt-type functions coming from setting z = 1 in this definition. We find some of the spt-crank-type functions to have interesting representations as single...

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