1 THE CHERRY TRANSFORM AND ITS RELATIONSHIP WITH A SINGULAR STURM-LIOUVILLE PROBLEM By H.M. SRIVASTAVA, VU KIM TUAN and S.B. YAKUBOVICH 1. Introduction In the usual notations, let D ν (z ) denote the parabolic cylinder function (or the Weber- Hermite function) of order (or index) ν , defined by (cf., e.g., [2, Vol. I, p. 267; Vol II, p. 117]) D ν (z )=2 ν 2 √ πe − z 2 4  1 Γ( 1−ν 2 ) 1 F 1  − ν 2 ; 1 2 ; z 2 2  − z √ 2 Γ(− ν 2 ) 1 F 1  1 − ν 2 ; 3 2 ; z 2 2  , (1.1) where 1 F 1 is the confluent hypergeometric function: 1 F 1 (a; c; z )= ∞  k=0 (a) k (c) k z k k! (1.2) with the Pochhammer symbol (a) k given by (a) k = Γ(a + k) Γ(a) = a(a + 1) ... (a + k − 1) (k =1, 2, 3,...). Indeed the function D ν (z ) is a solution of the differential equation: d 2 w dz 2 +  ν + 1 2 − z 2 4  w =0, (1.3) which appears frequently in applications; for example, in separation of variables of the following elliptic partial differential equation: Δu + f (z )u =0  u = u(x,y,z ); Δ = ∂ 2 ∂x 2 + ∂ 2 ∂y 2 + ∂ 2 ∂z 2  (1.4) in the parabolic cylindrical co-ordinates. In 1949 Cherry [1] considered an index transformation with D ν (z ) as its kernel and established that, if f ∈ L 1 (R) is of bounded variation in a neighborhood of x, then (see also [2, Vol. II, p. 124]) −2πi[f (x + 0) + f (x − 0)] =  − 1 2 +i∞ − 1 2 −i∞ e (ν + 1 2 ) πi 2 sin(νπ) dν  ∞ −∞  D ν (e πi 4 x)D −ν −1 (e − πi 4 t)