Abstract
In this paper, an attempt has been made to derive parameter derivatives of Jacobi polynomials with three variables on the simplex. They are obtained via parameter derivatives of the classical Jacobi polynomials P n ( α,β ) ( x ) with respect to their parameters. The study is motivated by the expansions of parameter derivatives which are obtained for the classical Jacobi polynomials over the interval (−1, 1), Jacobi polynomials with two variables on the triangle and some other families of orthogonal polynomials in two variables.
Highlights
In literature, there are many studies on the parameter derivatives of various special functions which find applications in applied mathematics, several branches of mathematics, mathematical and theoretical physics
[13, 14], author has studied the derivative of the Legendre function of the first kind, with respect to its degree Q, [wPQ (z) / wQ ]Q n (n ), and has obtained its some representations, which are encountered in some problems such as in the general theory of relativity and in solving some boundary value problems of potential theory, of electromagnetism and of heat conduction in solids
J0 for orthogonal polynomials with two variables, with O being a parameter and 0 d k d n; n 0 and has given parameter derivative representations for Jacobi polynomials with two variables on the triangle and some families of orthogonal polynomials in two variables by using the expansions (2) and (3) given for Jacobi polynomials with one variable
Summary
There are many studies on the parameter derivatives of various special functions which find applications in applied mathematics, several branches of mathematics, mathematical and theoretical physics. The derivatives of the associated Legendre function of the first kind with respect to its order and its degree (with applications to the construction of the associated Legendre function of the second kind of integer degree and order) have been obtained in [15, 16]. These derivatives of the associated Legendre function arise in solutions of some problems of heat conduction, theoretical acoustics and other branches of theoretical physics. To obtain parameter derivative representations for orthogonal polynomials with one variable, a general method has been investigated in the following form. ¦ [\ (D n) \ (D )] n 1 1 k 0D k (see ([1], ? 6.3 ), [7])
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