We give a combinatorial interpretation for the positive moments of the values at the edge of the critical strip of the -functions of modular forms of and . We deduce some results about the asymptotics of these moments. We extend this interpretation to the moments twisted by the eigenvalues of Hecke operators.
Let pₙ/qₙ = [a₀;a₁,...,aₙ] be the n-th convergent of the continued fraction expansion of [a₀;a₁,a₂,...]. Leaping convergents are those of every r-th convergent (n = 0,1,2,...) for fixed integers r and i with r ≥ 2 and i = 0,1,...,r-1. The leaping convergents for the e-type Hurwitz continued fractions have been studied. In special, recurrence relations and explicit forms of such leaping convergents have been treated. In this paper, we consider recurrence relations and explicit forms of the leaping...
Denote the n-th convergent of the continued fraction by pₙ/qₙ = [a₀;a₁,...,aₙ]. We give some explicit forms of leaping convergents of Tasoev continued fractions. For instance, [0;ua,ua²,ua³,...] is one of the typical types of Tasoev continued fractions. Leaping convergents are of the form (n=0,1,2,...) for fixed integers r ≥ 2 and 0 ≤ i ≤ r-1.
Let p > 3 be a prime, and let Rₚ be the set of rational numbers whose denominator is not divisible by p. Let Pₙ(x) be the Legendre polynomials. In this paper we mainly show that for m,n,t ∈ Rₚ with m≢ 0 (mod p), and , where (a/p) is the Legendre symbol and [x] is the greatest integer function. As an application we solve some conjectures of Z. W. Sun and the author concerning , where m is an integer not divisible by p.