AN EXTENSION OF THE BETA FUNCTION EXPRESSED AS A COMBINATION OF CONFLUENT HYPERGEOMETRIC FUNCTIONS

Recently several authors have extended the Beta function by using its integral representation. However, in many cases no expression of these extended functions in terms of classic special functions is known. In the present paper, we introduce a further extension by defining a family of functions G(r,s) : R-+(*) -> C, with r, s epsilon C and R (r) > 0. For given r, s, we prove that this function satisfies a second-order linear differential equation with rational coefficients. Solving this ODE, we express G(r,s) as a combination of confluent hypergeometric functions. From this we deduce a new integral relation satisfied by Tricomi's function. We then investigate additional specific properties of G(r,1) which take the form of new non trivial integral relations involving exponential and error functions. We discuss the connection between G(r,1) and Stokes' first problem (or Rayleigh problem) in fluid mechanics which consists in determining the flow created by the movement of an infinitely long plate. For r epsilon 1/2 N*, we find additional relations between G(r,1) and Hermite polynomials. In view of these results, we believe the family of extended beta functions G(r,s) will find further applications in two directions: (i) for improving our knowledge of confluent hypergeometric functions and Tricomi's function, (ii) and for engineering and physics problems.