By Ching-Li Chai (auth.), David V. Chudnovsky, Gregory V. Chudnovsky, Harvey Cohn, Melvyn B. Nathanson (eds.)

ISBN-10: 0387976701

ISBN-13: 9780387976709

ISBN-10: 1475741588

ISBN-13: 9781475741582

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**Extra resources for Number Theory: New York Seminar 1989–1990**

**Example text**

N). Note that in ratios of the corresponding Hankel determinants there inevitably appear values of a-functions quadratic in n - a startling difference from the usual basic hypergeometrie functions. To derive the expression of the generating function of Cn = a(u + nv + e}/{a(u + nv)a(e)} . e"+I1", we use the q-expansions given above: 2W l e -~z~ a(x + 1/) ,.. = -i. {X + 1 + Y + 1 + 2 X-I Y - 1 where X = exp{;;;;-x}, Y = exp{i;;;;-y}. Y. Y. Chudnovsky Specializing x = nu + v, y = e, we can look at the foUowing sequence: u(nu+v+e) .

1))lj~ol}-1/2. = = = Here we denote AO A, tPn-m+i(A;) Pn-m+i(A;) for i 0, ... ,1 and tPn-m+;(A;) qn-m+i(A;) for i 1 + 1, ... ,1 + m. The Hankel determinants corresponding to djJ(l)(A) are transformed similarly. For example, let D~l) be the Hankel determinant for the function 1(1)(,),), corresponding to the addition of k zeros: Al, ... ,Ak to I(A). Then = We can apply these formulas to a problem of expressing the continued fraction expansion of aseries, whose coefficients are finite differences of coefficients of a given series.

Dt ,-n-1 . V. v. Chudnovsky In particular we have a formal identity: (+00 Jo o 00 {L c~ t"}en. dt = p-1 . pt • n=O L00 c"p-". n::=O ReInark 2. _ q(nu) (-1 },,-1 . ·21··· (n-1! () 2' q u" This closed form expression for the determinant of derivatives in Weierstrass ~ function, belongs to Kiepert, though it is derived explicitly in [34J. These identities can be used to evaluate several classes of continued fractions. Let us start in a more general setting. ;,(") (t» to determine the continued fradion expansion of F(xlt) = 00 L n=O ';'(")(t) x"+1 .

### Number Theory: New York Seminar 1989–1990 by Ching-Li Chai (auth.), David V. Chudnovsky, Gregory V. Chudnovsky, Harvey Cohn, Melvyn B. Nathanson (eds.)

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