VKiryakova_fcaa113

TRANSMUTATION METHOD FOR SOLVING . . . 307[G m,np,q (σ) = G m,np,q σ∣ a ]1, . . . , a pb 1 , . . . , b q= 1 ∫2πiLwhere the integrand in (22) has the structurem∏ ∏Γ(b k − s) n Γ(1 − a j + s)G m,np,q (s) =k=1q∏k=m+1j=1Γ(1 − b k + s)p∏j=n+1G m,np,q (s)σ s ds, (22),Γ(a j − s)L is a suitable contour in C; the orders (m, n; p, q) are nonnegative integerssuch that 0 ≤ m ≤ q, 0 ≤ n ≤ q; and the parameters a j , j = 1, . . . , p, b k , k =1, . . . , q are arbitrary complex numbers such that(b k + l) ≠ (a j − l ′ − 1); l, l ′ = 0, 1, 2, . . . ; j = 1, . . . , p, k = 1, . . . , q.For further extensions of the generalized fractional integrals (21), involvingthe so-called Fox’s H-function, see for example in Kiryakova [15],Ch.5; [18], etc.Meijers’ G-functions are useful tools because almost all the special functionsof mathematical physics, as well as the basic elementary functions, canbe represented as G-functions. Especially, the generalized hypergeometricfunctions (ghf-s) p F q (s) which denotations we shall need in this paper, arethe most often used G-functions (see more in [12], Vol.1; [20]; [15], App.):pF q (a 1 , . . . , a p ; b 1 , . . . , b q ; s) ==q∏Γ(b j )j=1p∏Γ(a j )j=1∞∑k=0[G 1,pp,q+1(a 1 ) k . . . (a p ) k s k(b 1 ) k . . . (b q ) k k!−s∣]1 − a 1 , . . . , 1 − a p, (23)0, 1 − b 1 , . . . , 1 − b qwhere p ≤ q and s ∈ C, or p = q + 1 and |s| < 1, and the Pochhammersymbol denotes (a) 0 = 1, (a) k = Γ(a + k)/Γ(a) = a(a + 1) . . . (a + k − 1).Examples of ghf-s (23), discussed here, are the (normalized) Bessel functionsj ν (x) = 0 F 1 (−; ν + 1; −(x/2) 2 ), (9); the generalizes trigonometric andhyperbolic functions; the hyper-Bessel functions, the Gauss function, etc.Our generalized operators of integration of fractional multi-order (21)are natural extension of the fractional integration operators in the classical