scholarly journals The Euler Series Transformation and the Binomial Identities of Ljunggren, Munarini and Simons

Integers ◽  
2010 ◽  
Vol 10 (3) ◽  
Author(s):  
Khristo N. Boyadzhiev
Keyword(s):  
Transformation Formula ◽  
Series Transformation ◽  
Euler Series ◽  
Binomial Identities

AbstractWe point out that the curious identity of Simons follows immediately from Euler's series transformation formula and also from an identity due to Ljunggren. We also mention its relation to Legendre's polynomials. At the end we use the generalized Euler series transformation to obtain two recent binomial identities of Munarini.

Generalized Euler series transformation applied to halfwidth and shift of molecular spectral lines calculation

2006 ◽  
Author(s):  
A. D. Bykov ◽  
N. N. Lavrentieva ◽  
T. P. Mishina
Keyword(s):  
Spectral Lines ◽  
Series Transformation ◽  
Euler Series

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 series transformation formula and related polynomials

2005 ◽  
Vol 2005 (23) ◽  
pp. 3849-3866 ◽  
Author(s):  
Khristo N. Boyadzhiev
Keyword(s):  
Power Series ◽  
Gamma Function ◽  
Asymptotic Series ◽  
Transformation Formula ◽  
Bell Polynomials ◽  
New Class ◽  
Series Transformation ◽  
Lerch Transcendent ◽  
Series Of Functions ◽  
Case Based

We present a formula that turns power series into series of functions. This formula serves two purposes: first, it helps to evaluate some power series in a closed form; second, it transforms certain power series into asymptotic series. For example, we find the asymptotic expansions forλ>0of the incomplete gamma functionγ(λ,x)and of the Lerch transcendentΦ(x,s,λ). In one particular case, our formula reduces to a series transformation formula which appears in the works of Ramanujan and is related to the exponential (or Bell) polynomials. Another particular case, based on the geometric series, gives rise to a new class of polynomials called geometric polynomials.

scholarly journals TWO SUPERCONGRUENCES RELATED TO MULTIPLE HARMONIC SUMS

2021 ◽  
pp. 1-11
Author(s):  
ROBERTO TAURASO
Keyword(s):  
Binomial Identities

Abstract Let p be a prime and let x be a p-adic integer. We prove two supercongruences for truncated series of the form $$\begin{align*}\sum_{k=1}^{p-1} \frac{(x)_k}{(1)_k}\cdot \frac{1}{k}\sum_{1\le j_1\le\cdots\le j_r\le k}\frac{1}{j_1^{}\cdots j_r^{}}\quad\mbox{and}\quad \sum_{k=1}^{p-1} \frac{(x)_k(1-x)_k}{(1)_k^2}\cdot \frac{1}{k}\sum_{1\le j_1\le\cdots\le j_r\le k}\frac{1}{j_1^{2}\cdots j_r^{2}}\end{align*}$$ which generalise previous results. We also establish q-analogues of two binomial identities.

A transformation formula in the theory of partitions

1944 ◽  
Vol 11 (4) ◽  
pp. 873-887 ◽  
Author(s):  
Lowell Schoenfeld
Keyword(s):  
Transformation Formula

A transformation formula for Grothendieck residues and some of its applications

1989 ◽  
Vol 29 (3) ◽  
pp. 495-499
Author(s):  
A. M. Kytmanov
Keyword(s):  
Transformation Formula
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