Abstract
Several mathematicians have extensively investigated polynomials, their extensions, and their applications in various other research areas for a decade. Our paper aims to introduce another such polynomial, namely, Laguerre-based generalized Humbert polynomial, and investigate its properties. In particular, it derives elementary identities, recursive differential relations, additional symmetry identities, and implicit summation formulas.
Highlights
In all the given definitions, let C, R, R+, and N be the sets of complex numbers, real numbers, positive real numbers, and natural numbers, respectively
Its generalization is given by the two variable Laguerre polynomial. e twovariable Laguerre polynomial Ln(x, y) is defined by the following generating function: eytC0(xt)
Generalization of two variables of all the above polynomials and many more was given by Djordjevic in the form 1 − 2(x + y)t + tm(2xy + 1)− α Gαn,m(x, y)tn, n 0 (13)
Summary
In all the given definitions, let C, R, R+ , and N be the sets of complex numbers, real numbers, positive real numbers, and natural numbers, respectively. Its generalization is given by the two variable Laguerre polynomial. E twovariable Laguerre polynomial Ln(x, y) is defined by the following generating function (see [5,6,7,8]): eytC0(xt). E following generating function gives the extension of equations (6) and (7): 2xt + t2− ]. Generalization of two variables of all the above polynomials and many more was given by Djordjevic (see [10]) in the form 1 − 2(x + y)t + tm(2xy + 1)− α Gαn,m(x, y)tn, n 0. E three-variable Hermite–Laguerre polynomial HLn(x, y, z) is defined by the following generating function (see [6]): eyt+zt C0(xt). E Laguerre-based generalized Humbert polynomials of order ], denoted by LGυn,m(a, b, c; x, y, z), is defined by the following generating function:. Using (1), (4), and (13) in (30) and rearranging the equations, we get the desired result in (28)
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