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) '

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