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The present work deals with the mathematical investigation of the product formulas and several q-Laplace type integral transforms of certain q-Humbert functions. In our investigation, the qL2-transform and qL2-transform of certain q2-Humbert functions are considered. Several useful special cases have been deduced as applications of main results. [ABSTRACT FROM AUTHOR]

Radial free harmonic wave propagation is considered for the investigation in cylindrical shells made of orthotropic human bones, transversely isotropic hexagonal crystals and simply isotropic materials under the effect of magnetic field and hydrostatic state of stress. Numerical investigations are performed for the frequency equation through 2D and 3D graphical visuals for the impact of rotation, hydrostatic stress, axial magnetic field, inner and outer radii of the cylindrical shell on the radial phase velocity of the wave. A comparative study has been developed for femoral, tibia, mandibular buccal and lingua bones, Mg, Co, Cd, Zn, beryl and ice hexagonal crystals. [ABSTRACT FROM AUTHOR]

The Gould–Hopper–Laguerre–Sheffer matrix polynomials were initially studied using operational methods, but in this paper, we investigate them using matrix techniques. By leveraging properties of Pascal functionals and Wronskian matrices, we derive several identities for these polynomials, including recurrence relations. It is highlighted that these identities, acquired via matrix techniques, are distinct from the ones obtained when using operational methods. [ABSTRACT FROM AUTHOR]

This work deals with the mathematical inspection of a hybrid family of the degenerate polynomials of the Apostol-type. The inclusion of the derivation of few series expansion formulas, explicit representations and difference equations for this hybrid family brings a novelty to the existing literature. Moreover, certain connection formulas and several novel identities for these polynomials are established and investigated. The graphical representations of certain degenerate polynomials are explored and several new interesting pattern of the zeros are observed. [ABSTRACT FROM AUTHOR]

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]

The quantum groups nowadays attract a considerable interest of mathematicians and physicists. The theory of q-special functions has received a group-theoretic interpretation using the techniques of quantum groups and quantum algebras. This paper focuses on introducing the q-Tricomi functions and 2D q-Tricomi functions through the generating function and series expansion and for the first time establishing a connecting relation between the q-Tricomi and q-Bessel functions. The behavior of these functions is described through shapes, and the contrast between them is observed using mathematical software. Further, the problem of framing the q-Tricomi and 2D q-Tricomi functions in the context of the irreducible representation (ω) of the two-dimensional quantum algebra ℰq(2) is addressed, and certain relations involving these functions are obtained. 2-Variable 1-parameter q-Tricomi functions and their relationship with the 2-variable 1-parameter q-Bessel functions are also explored. [ABSTRACT FROM AUTHOR]

The N-transform (so-called natural transform) of a function, which may combine Laplace and Sumudu transforms, has been introduced and investigated. Recently $q$-analogues of the ${\bf N}$-transform has been presented and studied. Among $q$-extensions of a number of polynomials and functions, the $q$-Humbert functions and some of their interesting identities and properties have recently been introduced and provided. In this paper, we aim to present $q$-natural transforms of a finite product of the $q$-Humbert functions. Some particular cases of our main identities are also considered. [ABSTRACT FROM AUTHOR]

The present paper intends to introduce the hybrid form of q-special polynomials, namely q-Hermite-Appell polynomials by means of generating function and series definition. Some significant properties of q-Hermite-Appell polynomials such as determinant definitions, q-recurrence relations and q-difference equations are established. Examples providing the corresponding results for certain members belonging to this q-Hermite-Appell family are considered. In addition, graphs of certain q-special polynomials are demonstrated using computer experiment. Thereafter, distribution of zeros of these q-special polynomials is displayed. [ABSTRACT FROM AUTHOR]