PROBABILISTIC DERIVATION OF A q-POLYNOMIALS TRANSFORMATION FORMULA

2013 ◽  
Vol 16 (02) ◽  
pp. 1350017 ◽  
Author(s):  
MINGJIN WANG
Keyword(s):  
Convergence Theorem ◽  
Summation Formula ◽  
Transformation Formula ◽  
Dominated Convergence ◽  
Transformation Formulas

In this paper, we use the distribution W(x; q) and the Lebesgue's dominated convergence theorem to give a probabilistic derivation of a transformation formula for the Al-Salam-Carlitz polynomials. Some applications of the transformation are also given, which include an extension of the Heine's 2ϕ1 transformation formulas and an extension of the Bailey-Daum summation formula.

scholarly journals Some q-Supercongruences from Transformation Formulas for Basic Hypergeometric Series

2020 ◽  
Author(s):  
Victor J. W. Guo ◽  
Michael J. Schlosser
Keyword(s):  
Hypergeometric Series ◽  
Basic Hypergeometric Series ◽  
Double Series ◽  
Summation Formula ◽  
Transformation Formula ◽  
Quadratic Transformation ◽  
Ultraspherical Polynomials ◽  
Higher Power ◽  
Series Transformation ◽  
Transformation Formulas

AbstractSeveral new q-supercongruences are obtained using transformation formulas for basic hypergeometric series, together with various techniques such as suitably combining terms, and creative microscoping, a method recently developed by the first author in collaboration with Zudilin. More concretely, the results in this paper include q-analogues of supercongruences (referring to p-adic identities remaining valid for some higher power of p) established by Long, by Long and Ramakrishna, and several other q-supercongruences. The six basic hypergeometric transformation formulas which are made use of are Watson’s transformation, a quadratic transformation of Rahman, a cubic transformation of Gasper and Rahman, a quartic transformation of Gasper and Rahman, a double series transformation of Ismail, Rahman and Suslov, and a new transformation formula for a nonterminating very-well-poised $${}_{12}\phi _{11}$$ 12 ϕ 11 series. Also, the nonterminating q-Dixon summation formula is used. A special case of the new $${}_{12}\phi _{11}$$ 12 ϕ 11 transformation formula is further utilized to obtain a generalization of Rogers’ linearization formula for the continuous q-ultraspherical polynomials.

scholarly journals A dominated convergence theorem for Eisenstein series

2021 ◽  
Author(s):  
Johann Franke
Keyword(s):  
Fourier Series ◽  
Dirichlet Series ◽  
Eisenstein Series ◽  
Modular Forms ◽  
Convergence Theorem ◽  
Rational Functions ◽  
New Approach ◽  
Series Representations ◽  
Dominated Convergence ◽  
Generalized Dirichlet

AbstractBased on the new approach to modular forms presented in [6] that uses rational functions, we prove a dominated convergence theorem for certain modular forms in the Eisenstein space. It states that certain rearrangements of the Fourier series will converge very fast near the cusp $$\tau = 0$$ τ = 0 . As an application, we consider L-functions associated to products of Eisenstein series and present natural generalized Dirichlet series representations that converge in an expanded half plane.

scholarly journals On Fuzzy Henstock-Kurzweil-Stieltjes-♦-Double Integral on Time scales

2021 ◽  
Vol 2 (2) ◽  
pp. 38-49
Author(s):  
David AFARIOGUN ◽  
Adesanmi MOGBADEMU ◽  
Hallowed OLAOLUWA
Keyword(s):  
Uniform Convergence ◽  
Time Scales ◽  
Convergence Theorem ◽  
Monotone Convergence ◽  
Double Integral ◽  
Monotone Convergence Theorem ◽  
Integrable Functions ◽  
Dominated Convergence

We introduce and study some properties of fuzzy Henstock-Kurzweil-Stietljes-$ \Diamond $-double integral on time scales. Also, we state and prove the uniform convergence theorem, monotone convergence theorem and dominated convergence theorem for the fuzzy Henstock-Kurzweil Stieltjes-$\Diamond$-double integrable functions on time scales.

scholarly journals A DOMINATED CONVERGENCE THEOREM IN THE K-H INTEGRAL

2003 ◽  
Vol 7 (3) ◽  
pp. 507-512
Author(s):  
Jitan Lu ◽  
Peng-Yee Lee
Keyword(s):  
Convergence Theorem ◽  
Dominated Convergence

scholarly journals Fatou's Lemma and Lebesgue's convergence theorem for measures

2000 ◽  
Vol 13 (2) ◽  
pp. 137-146 ◽  
Author(s):  
Onésimo Hernández-Lerma ◽  
Jean B. Lasserre
Keyword(s):  
Asymptotic Behavior ◽  
Banach Spaces ◽  
General Result ◽  
Convergence Theorem ◽  
Convergence Theorems ◽  
Vector Lattices ◽  
Fatou’S Lemma ◽  
Dominated Convergence ◽  
Special Case ◽  
Fatou's Lemma

Analogues of Fatou's Lemma and Lebesgue's convergence theorems are established for ∫fdμn when {μn} is a sequence of measures. A “generalized” Dominated Convergence Theorem is also proved for the asymptotic behavior of ∫fndμn and the latter is shown to be a special case of a more general result established in vector lattices and related to the Dunford-Pettis property in Banach spaces.

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