scholarly journals -Analogues of Symbolic Operators

2013 ◽  
Vol 2013 ◽  
pp. 1-6 ◽  
Author(s):  
Michael J. Dancs ◽  
Tian-Xiao He
Keyword(s):  
Linear Operators ◽  
Derivative Operator ◽  
Symbolic Summation ◽  
Series Transformation ◽  
Transformation Formulas ◽  
Symbolic Substitution

Here presented are -extensions of several linear operators including a novel -analogue of the derivative operator . Some -analogues of the symbolic substitution rules given by He et al., 2007, are obtained. As sample applications, we show how these -substitution rules may be used to construct symbolic summation and series transformation formulas, including -analogues of the classical Euler transformations for accelerating the convergence of alternating series.

scholarly journals Power and Chebyshev Series Transformation Formulas with Applications to Solving Ordinary Differential Equations via the Fröbenius and Taylor’s methods

OALib ◽  
2021 ◽  
Vol 08 (02) ◽  
pp. 1-19
Author(s):  
Hippolyte Nyengeri ◽  
Rénovat Nizigiyimana ◽  
Jean-Pierre Mutankana ◽  
Henry Bayaga ◽  
Ferdinand Bayubahe
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Chebyshev Series ◽  
Series Transformation ◽  
Transformation Formulas

scholarly journals On a Generalized Convolution Operator

Symmetry ◽  
2021 ◽  
Vol 13 (11) ◽  
pp. 2141
Author(s):  
Poonam Sharma ◽  
Ravinder Krishna Raina ◽  
Janusz Sokół
Keyword(s):  
Fractional Derivative ◽  
Convolution Operator ◽  
Analytic Functions ◽  
Linear Operators ◽  
Generalized Convolution ◽  
Derivative Operator ◽  
Special Cases ◽  
Lerch Zeta Function ◽  
The One ◽  
Appell Function

Recently in the paper [Mediterr. J. Math. 2016, 13, 1535–1553], the authors introduced and studied a new operator which was defined as a convolution of the three popular linear operators, namely the Sǎlǎgean operator, the Ruscheweyh operator and a fractional derivative operator. In the present paper, we consider an operator which is a convolution operator of only two linear operators (with lesser restricted parameters) that yield various well-known operators, defined by a symmetric way, including the one studied in the above-mentioned paper. Several results on the subordination of analytic functions to this operator (defined below) are investigated. Some of the results presented are shown to involve the familiar Appell function and Hurwitz–Lerch Zeta function. Special cases and interesting consequences being in symmetry of our main results are also mentioned.

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 Differential Subordination and Superordination for a Generalized Differential Operator Involving Mittag-Leffler Function

2021 ◽  
Vol 45 (5) ◽  
pp. 699-708
Author(s):  
SUHILA ELHADDAD ◽  
◽  
MASLINA DARUS
Keyword(s):  
Differential Operator ◽  
Differential Subordination ◽  
Linear Operators ◽  
Derivative Operator ◽  
Sandwich Type ◽  
Differential Subordination And Superordination ◽  
Generalized Differential

Abstract. Owning to the importance and great interest of linear operators, a generalisation of linear derivative operator He m δ,p(α, β, a1, b1)f(z) is newly introduced in this study. The main objective of this paper is to investigate various subordination and superordination related to the aforementioned generalised linear derivative operator. Additionally, the resultant sandwich-type of this operator is also considered.

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