Integral equations for special functions of Heun class

Abstract

Abstract. Integral equations equivalent to boundary problems for the differen-tial equations of Heun class are studied. A new method of their derivation isproposed which is not as general as the known ones but is efficient in creatingintegral equations of a certain type. A number of new integral equations are ob-tained, some related to the double-confluent Heun equation. Formal calculationsat the first stage are followed by rigorous formulations. 1. IntroductionThe well-known special functions of mathematical physics, Bessel functions, hyper-geometric functions, etc., are defined as solutions of linear ordinary second-order dif-ferential equations with polynomial coefficients. Below we call any solution of suchequations a special function. The special functions can be expressed by means of in-tegral representations in terms of elementary functions. These representations are ofparticular importance for further study of additional characteristics of these functions:recurrence relations, asymptotic expansions, etc.There is no hope of finding such representations for all special functions in ourmore general sense; however, there exist more sophisticated integral relations andintegral equations which do succeed. Our main goal in this article is to deduce severalsimple integral equations for some special functions, namely of Heun type and thecorresponding confluent cases [4, 10]. These functions usually arise as eigenfunctionsof an eigenvalue problem associated with the initial differential equation.Several of our results are known. The main tools used previously for their derivationare either utilization of some integral transforms [6] or separation of variables in anauxiliary partial differential equation (pde) [1, 2, 5, 7-9]. Our approach is based onthe polynomial structure of the original equation, and it permits us to obtain newrelations as well as to give a new derivation of the old ones in a unique manner.On the other hand, it does not give the large variety of relations depending on anadditional parameter obtained by separation of variables of the mentioned pde. Infact these two methods are complementary to each other.2. Major definitionsBy a Heun equation we mean a second-order linear homogeneous differential equationwith four regular singular points. With the help of a linear transformation of theindependent variable these points can be located at z