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The present work deals with the mathematical investigation of some generalizations of Bessel functions. The main motive of this paper is to show that the generating function can be employed efficiently to obtain certain results for special functions. The complex form of Bessel functions is introduced by means of the generating function. Certain enthralling properties for complex Bessel functions are investigated using the generating function method. By considering separately the real and the imaginary part of complex Bessel functions, we get respectively cosine-Bessel functions and sine-Bessel functions for which several novel identities and Jacobi–Anger expansions are established. Also, the generating function of degenerate Bessel functions is investigated and certain novel identities are obtained for them. A hybrid form of degenerate Bessel functions, namely, of degenerate Fubini–Bessel functions, is constructed using the replacement technique. Finally, the explicit forms of the real and the imaginary part of complex Bessel functions are established by a hypergeometric approach. [ABSTRACT FROM AUTHOR]

In this paper, a summation theorem for the Clausen series is derived. Further, a reduction formula is obtained for the Kampé de Fériet double hypergeometric function. Some special cases are given as applications. A generalization of the reduction and linear transformation formulas is also given in the form of the general double series identity. [ABSTRACT FROM AUTHOR]

In this paper, some infinite product representations of certain transcendental functions (whose all possible zeros and n-th order differential coefficient at the point x = 0, can be calculated easily) are obtained by using the theory of polynomial equations. This approach is novel as it is different from the approaches used by other authors. [ABSTRACT FROM AUTHOR]

In this article, certain identities involving the Sheffer-Appell polynomials are established using the properties and relationships between the Pascal functional matrices and Wronskian matrices. Further, the corresponding results for certain members belonging to these families are also considered as illustrations. [ABSTRACT FROM AUTHOR]

In this work, the generalized family of Hermite–Sheffer polynomials is introduced by using Euler's integral and operational rules. Furthermore, some properties are established. In particular, generating function and determinant definition for the generalized Hermite–Sheffer polynomials are obtained. Some examples are also considered and their corresponding results are established. [ABSTRACT FROM AUTHOR]

The use of integral transforms and operational methods is a fairly useful tool to deal with new families of special polynomials. In this article, the extended Laguerre–Gould–Hopper–Appell polynomials are introduced by means of generating function and determinant definition. Their quasi-monomial properties are also established. Examples of some members belonging to the extended Laguerre–Gould–Hopper–Appell polynomials are considered, and their contour and 3-D plots are drawn. [ABSTRACT FROM AUTHOR]

In this work, the 2D-Appell and the Bessel polynomials are combined to introduce the family of the 2D-Appell-Bessel polynomials. The generating function, quasi-monomial properties, series definition and determinant form of these polynomials are established. Examples of some members belonging to this family are considered. The graphs of some hybrid special polynomials are also drawn for suitable values of the indices. [ABSTRACT FROM AUTHOR]

The intended objective of this paper is to introduce a new class of polynomials, namely the extended Laguerre-Gould-Hopper-Sheffer polynomials. The generating function and operational rule are derived by making use of integral transform. Their quasi-monomial properties and determinant forms are also established. Examples of certain members belonging to the extended Laguerre-Gould-Hopper-Sheffer polynomials are constructed and their corresponding results are established. [ABSTRACT FROM AUTHOR]

In this article, a method based on linear algebra approach is adopted to study the hybrid Sheffer–Appell polynomials. The recursive formulas and differential equation for these polynomials are derived by using the properties and relationships between the Pascal functional matrices and the Wronskian matrices. The corresponding results for some mixed type special polynomials are also obtained. [ABSTRACT FROM AUTHOR]

In this article, the discrete Bessel and Tricomi convolutions of the Laguerre-Gould-Hopper polynomials are introduced. Some important properties including recurrence relations and operational representations of these families are established. For suitable values of indices and variables, the Laguerre-Gould-Hopper polynomials yield several special polynomials. Consequently, results for the discrete Bessel and Tricomi convolutions of the corresponding special polynomials are also obtained. [ABSTRACT FROM AUTHOR]

This article is written with the aim to establish an equivalence between three different approaches related to special polynomial sequences. A result is proved showing the equivalence of the representation of a Lie algebra, quasi-monomiality and Lie algebra of endomorphism of vector spaces. The result is illustrated with the help of an example. [ABSTRACT FROM AUTHOR]

In this paper, a result for double index sequences is established by using a quadratic combination of four operators defined on a Lie algebra of endomorphisms of a vector space. This result is used to give new proofs of some properties of the Bessel and Tricomi functions of two indices. Further, this result is extended for multi-index sequences. [ABSTRACT FROM AUTHOR]

The representations of the Lie algebras generate in a natural way all known classical special polynomials. This allows one to greatly simplify the theory of orthogonal polynomials by expressing them in terms of the corresponding Lie algebra or Lie group. In this article, the problem of framing the 1-parameter 2D-Hermite polynomials (1P2DHP)(which are 2D orthogonal polynomials) into the context of the irreducible representationsandof the four-dimensional Lie algebrais considered. This approach stress the mathematical relevance of 2D-orthogonal polynomials and Lie algebras. Certain relations involving the 1P2DHPare obtained using the approach adopted by Miller. The linear differential operators serve as useful tools towards obtaining these relations. The analysis has been carried out by generalizing the formalism relevant to 1P2DHP. Certain examples involving 2D-Hermite polynomialsand Laguerre polynomialsare obtained as special cases. [ABSTRACT FROM AUTHOR]

In this paper, certain mixed special polynomial families associated with Appell sequences are introduced and their properties are established. Further, operational rules providing connections between these families and known special polynomials are established, which are used to derive the identities and results for the members of these new families. Determinantal definitions of the polynomials associated with Appell family are also derived. The approach presented is general. [ABSTRACT FROM AUTHOR]

Sine oculis homeobox homolog 1 (Six1) is a developmentally important transcription factor that regulates cellular proliferation, apoptosis, and dissemination during embryogenesis. Six1 overexpression as reported in multiple cancers modulates expression of a repertoire of its target genes causing an increase in proliferation, metastasis and survival of cancer cells. Six1 exists as a cell cycle regulated nuclear phosphoprotein and its cellular turnover is regulated by APC/C (Anaphase promoting complex / Cyclosome) complex mediated proteolysis. However, the kinases that regulate Six1 proteolysis have not been identified and the mechanistic details that cause its overproduction in various cancers are lacking. Here, we report that Six1 is a physiological GSK3β substrate. GSK3β interacts with Six1 and phosphorylates it at Ser221 within the conserved consensus sequence in its carboxy terminus. Using pharmacological inhibition, siRNA mediated knockdown and protein overexpression of GSK3β; we show that GSK3β regulates Six1 protein stability. Pulse chase analysis of Six1 revealed that GSK3β regulates its ubiquitin proteolysis such that Six1 phosphomimicking mutant (Six1S221E) for Ser221 site had dramatically increased half-life than its phosphodeficient (Six1S221A) and wild type variants. Furthermore, we demonstrate that GSK3β rescues Six1 from APC dependent proteolysis by regulating its binding with APC/C co-activator protein Cdh1. Importantly, strong positive correlation exists between GSK3β and Six1 protein levels throughout the cell cycle and in multiple cancers indicating that GSK3β activation may in part contribute to Six1 overproduction in a subset of human cancers. • Six1 is a GSK3β Substrate. • GSK3β phosphorylates Six1 at Ser221 in its carboxy terminus. • GSK3β regulates Six1 turnover by regulating its binding with APC adaptor protein Cdh1. • GSK3β activity strictly follows Six1 protein levels during cell cycle progression [ABSTRACT FROM AUTHOR]