Some decomposition formulas of generalized hypergeometric functions and formulas of an analytic continuation of the Clausen function

TL;DR

This paper develops new decomposition formulas for generalized hypergeometric functions and derives analytic continuation formulas for the Clausen function, connecting it with the well-studied Appell hypergeometric functions.

Contribution

It introduces novel symbolic expansion formulas for hypergeometric functions and provides an analytic continuation framework for the Clausen function using these formulas.

Findings

Derived new expansion formulas for generalized hypergeometric functions.

Established analytic continuation formulas for the Clausen function.

Connected Clausen function continuation with Appell functions.

Abstract

In this paper, using similar symbolical method of Burchnall and Chaundy formulas of expansion for the generalized hypergeometric function were constructed. By means of the found formulas of expansion the formulas of an analytic continuation for hypergeometric function of Clausen is defined. The obtained formulas of an analytic continuation express known hypergeometric Appell function $ F_2 ({a;b_1, b_2 ;c_1, c_2 ;x,y}) $ which theory is well studied.