A class of integral equations involving the H function as kernel

AbstractIntegral equations of the type (∗) ∫0xK(xy)f(y)(dyy)=g(x),x≧ where g is given and f is an unknown function to be determined, can be reduced fairly easily to the form (∗∗) ∫u∞K1(t−u)f1(t)dt=g1(u),u≧0In the present paper the author shows how the systematic use of the theory of the Mellin transform leads to a simple procedure by means of which this class of integral equations may be solved. The technique, which presupposes the existence of the Euler transform of the kernel as well as the Mellin transform of the Euler transform, is illustrated here by obtaining a solution of the integral equation (∗∗∗) ∫y∞(x−y)−αHp,qm,n[x−]f(x)dx=g(y),y≧0 [ζ] denotes the H-function of C. Fox [2]. Since a large variety of functions that occur frequently in problems of analysis and mathematical physics are only specialized or limiting forms of the kernel used in (∗∗∗), the inversion problem discussed in this paper may prove to be of general interest. Several particular cases of the main result are discussed systematically