Abstract
The main aim of this paper is to define and study of a new Horn’s matrix function, say, the p and q-Horn’s matrix function of two complex variables. The radius of regularity on this function is given when the positive integers p and q are greater than one, an integral representation of pHq 2 is obtained, recurrence relations are established. Finally, we obtain a higher order partial differential equation satisfied by the p and q-Horn’s matrix function.
Highlights
We obtain a higher order partial differential equation satisfied by the p and q-Horn’s matrix function
Many special functions encountered in mathematical physics, theoretical physics, engineering and probability theory are special cases of hypergeometric functions [1]
In [6,7], the hypergeometric matrix function has been introduced as a matrix power series and an integral representation
Summary
Many special functions encountered in mathematical physics, theoretical physics, engineering and probability theory are special cases of hypergeometric functions [1]. In [6,7], the hypergeometric matrix function has been introduced as a matrix power series and an integral representation. Upadhyaya and Dhami have earlier studied the generalized Horn’s functions of matrix arguments with real positive definite matrices as arguments [10] and this function H7 [11], while the author has earlier studied the Horn’s matrix function H2 of two complex variables under differential operators [7]. Our purpose here is to introduce and study an extension of the matrix functions of two variables. This paper is organized as follows: Section 2 contains the definition of the p and q-Horn’s matrix function of two variables, its radius of regularity and integral relation of the p and q-Horn’s matrix function is given. The gamma matrix function (P) and the beta matrix function B(P,Q) have been defined in [16], as follows (P) e tt P I dt; t P I e(P I )ln t 0
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