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Generalized Cesàro operators on certain function spaces

Sunanda Naik (2010)

Annales Polonici Mathematici

Motivated by some recent results by Li and Stević, in this paper we prove that a two-parameter family of Cesàro averaging operators b , c is bounded on the Dirichlet spaces p , a . We also give a short and direct proof of boundedness of b , c on the Hardy space H p for 1 < p < ∞.

Generalized hermite polynomials obtained by embeddings of the q-Heisenberg algebra

Joachim Seifert (1997)

Banach Center Publications

Several ways to embed q-deformed versions of the Heisenberg algebra into the classical algebra itself are presented. By combination of those embeddings it becomes possible to transform between q-phase-space and q-oscillator realizations of the q-Heisenberg algebra. Using these embeddings the corresponding Schrödinger equation can be expressed by various difference equations. The solutions for two physically relevant cases are found and expressed as Stieltjes Wigert polynomials.

Generalized Krawtchouk polynomials: New properties

Norris Sookoo (2000)

Archivum Mathematicum

Orthogonality conditions and recurrence relations are presented for generalized Krawtchouk polynomials. Coefficients are evaluated for the expansion of an arbitrary polynomial in terms of these polynomials and certain special values for generalized Krawtchouk polynomials are obtained. Summations of some of these polynomials and of certain products are also considered.

Generalized trigonometric functions in complex domain

Petr Girg, Lukáš Kotrla (2015)

Mathematica Bohemica

We study extension of p -trigonometric functions sin p and cos p to complex domain. For p = 4 , 6 , 8 , ⋯ , the function sin p satisfies the initial value problem which is equivalent to (*) - ( u ' ) p - 2 u ' ' - u p - 1 = 0 , u ( 0 ) = 0 , u ' ( 0 ) = 1 in ℝ . In our recent paper, Girg, Kotrla (2014), we showed that sin p ( x ) is a real analytic function for p = 4 , 6 , 8 , ⋯ on ( - π p / 2 , π p / 2 ) , where π p / 2 = ∫ 0 1 ( 1 - s p ) - 1 / p . This allows us to extend sin p to complex domain by its Maclaurin series convergent on the disc { z ∈ ℂ : | z | < π p / 2 } . The question is whether this extensions sin p ( z ) satisfies (*) in the sense of differential equations in complex domain. This interesting...

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