Arithmetic properties of the values of hypergeometric functions with different parameters

Abstract

The arithmetic properties of the values of the hypergeometric functions can in some cases be studied by the effective construction of a system of linear approximating forms. This construction was used for functions with several parameters in [I], where functions of the type in part (B) of the theorem below were studied for a(x) ~ 1 and b(x) ~ i. The arithmetic properties of the values of polylogarithms with several parameters were studied in [2, 3]. The estimates of linear forms obtained in this manner are more accurate than those obtained (in a more general situation) by means of the Siegel method (see Remark 1 below). The proof of the theorem given below is based on the effective construction of a system of linear approximating forms carried out in [4]. Here to prove each part of the theorem we use a special auxiliary function which plays the same role in the proof as the function ~(z) from Lemma 1 of [4]. THEOREM. Let I be an imaginary quadratic field or a field of rational numbers Q; a ( x ) = ( x + a , ) . . . (x+a~.); b ( x ) = ( x + ~ , ) . . . (x+~m); 5; a 1 . . . . . a r ; $1 . . . . . Sm; ~1 . . . . ,A t a r e r a t i o n a l number s ; a i ~ j , ~h,--~h~Z (i=1 . . . . . r; ] = t . . . . , m; ki, k ~ = t , . . . , t ; k ~ # k 2 ) ; ( x + ~ + ~ ) ( x + ~ k ) a ( x ) b ( x ) # O f o r x = t , 2, 3, . . . ; k = t . . . . . t; ~ I ~ ~r Furthermore, let ; , x -~ ~ l-[ ~ (~ + ~ ) ~ (x) (A) r (z) = ~'--,=o ~-1 b (x) '