The convergence behavior of q-continued fractions on the unit circle

In a previous paper, we showed the existence of an uncountable set of points on the unit circle at which the Rogers-Ramanujan continued fraction does not converge to a finite value. In this present paper, we generalise this result to a wider class of q-continued fractions, a class which includes the Rogers-Ramanujan continued fraction and the three Ramanujan-Selberg continued fractions. We show, for each q-continued fraction, G(q), in this class, that there is an uncountable set of points, Y _G , on the unit circle such that if y ? Y _G then G(y) does not converge to a finite value. We discuss the implications of our theorems for the convergence of other q-continued fractions, for example the Göllnitz-Gordon continued fraction, on the unit circle.     

Arithmetic of the Ramanujan-Göllnitz-Gordon continued fraction

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We extend the results of Chan and Huang [H.H. Chan, S.-S. Huang, On the Ramanujan-Göllnitz-Gordon continued fraction, Ramanujan J. 1 (1997) 75-90] and Vasuki, Srivatsa Kumar [K.R. Vasuki, B.R. Srivatsa Kumar, Certain identities for Ramanujan-Göllnitz-Gordon continued fraction, J. Comput. Appl. Math. 187 (2006) 87-95] to all odd primes p on the modular equations of the Ramanujan-Göllnitz-Gordon continued fraction v(τ) by computing the affine models of modular curves X(Γ) with Γ=Γ₁(8)∩Γ₀(16p). We then deduce the Kronecker congruence relations for these modular equations. Further, by showing that v(τ) is a modular unit over Z we give a new proof of the fact that the singular values of v(τ) are units at all imaginary quadratic arguments and obtain that they generate ray class fields modulo 8 over imaginary quadratic fields.

Video

For a video summary of this paper, please visit http://www.youtube.com/watch?v=FWdmYvdf5Jg.     

G-continued fractions for basic hypergeometric functions II

In this paper we apply a modification of a generalized Pringsheim's theorem to obtain a G-continued fraction expansion for the quotient of two contiguous basic hypergeometric functions in arbitrarily many variables. As an application we obtain a G-continued fraction extension of the Rogers-Ramanujan continued fraction.     

Tasoev's continued fractions and Rogers-Ramanujan continued fractions

We show some new variations on Tasoev's continued fractions , where the periodic parts include the exponentials in k instead of the polynomials in k. We also mention some relations with other kinds of continued fractions, in particular, with Rogers-Ramanujan continued fractions.     

The explicit formulas and evaluations of Ramanujan's theta-function ψ

We define two quotients of theta-function ψ depending on two positive real parameters. We then show how they are connected with two parameters of Dedekind eta-function, theta-function φ, and the Ramanujan-Weber class invariants. Explicit formulas for determining values of the theta-function ψ are derived, and several examples will be given. In addition, we give some applications of these parameters for the famous Rogers-Ramanujan continued fraction R(q), Ramanujan's cubic continued fraction G(q), and the modular j-invariant.     

Shrinking the period length of quasi-periodic continued fractions

Extending the work of Burger et al., here we show that every quasi-periodic simple continued fraction α can be transformed into a quasi-periodic non-simple continued fraction having period length one. Moreover, a certain kind of quasi-periodic non-simple continued fraction is equivalent to a quasi-periodic N-continued fraction. The results of this paper follow from arguments of Burger et al. but we apply our version to offer new continued fractions for certain classes of real numbers.     

Some Continued Fractions in Ramanujan’s Lost Notebook

In this paper we first give the value of a periodic continued fraction which was recorded incorrectly by Ramanujan on page 341 of his lost notebook. Next, we describe several pairs of equivalent continued fractions in which one is the odd part of the other. One of the results is for the Rogers-Ramanujan continued fraction which was recently proved by Berndt and Yee. Finally, using the Bauer-Muir transformation we prove the equivalence of two continued fractions. One was recorded on page 44 in Ramanujan’s lost notebook, and the other is found in the unorganized pages at the end of Ramanujan’s second notebook.This work was supported by Yonsei University Research Fund of 2003.     

A recurrence formula for leaping convergents of non-regular continued fractions

Given a continued fraction [a₀;a₁,a₂,…], p_n/q_n=[a₀;a₁,…,a_n] is called the n-th convergent for n=0,1,2,…. Leaping convergents are those of every r-th convergent p_rn+i/q_rn+i (n=0,1,2,…) for fixed integers r and i with r?2 and i=0,1,…,r-1. This leaping step r can be chosen as the length of period in the continued fraction. Elsner studied the leaping convergents p_3n+1/q_3n+1 for the continued fraction of and obtained some arithmetic properties. Komatsu studied those p_3n/q_3n for (s?2). He has also extended such results for some more general continued fractions. Such concepts have been generalized in the case of regular continued fractions. In this paper leaping convergents in the non-regular continued fractions are considered so that a more general three term relation is satisfied. Moreover, the leaping step r need not necessarily to equal the length of period. As one of applications a new recurrence formula for leaping convergents of Apery’s continued fraction of ζ(3) is shown.     

Some Theorems on the Rogers-Ramanujan Continued Fraction in Ramanujan's Lost Notebook

In his first two letters to G. H. Hardy and in his notebooks, Ramanujan recorded many theorems about the Rogers-Ramanujan continued fraction. In his lost notebook, he offered several further assertions. The purpose of this paper is to provide proofs for many of the claims about the Rogers-Ramanujan and generalized Rogers-Ramanujan continued fractions found in the lost notebook. These theorems involve, among other things, modular equations, transformations, zeros, and class invariants.     

Special values of L-functions by a Siegel-Weil-Kudla-Rallis formula

We study the arithmeticity of special values of L-functions attached to cuspforms which are Hecke eigenfunctions on hermitian quaternion groups Sp^∗(m,0) which form a reductive dual pair with G=O^∗(4n). For f₁ and f₂ two cuspforms on H, consider their theta liftings θf₁ and θf₂ on G. Then we compute a Rankin-Selberg type integral and obtain an integral representation of the standard L-function:

G〈θf₁⋅Es,θf₂〉=H〈f₁,f₂〉⋅Lstd(f₁,s).     

The theta group and the continued fraction expansion with even partial quotients

F. Schweiger introduced the continued fraction with even partial quotients. We will show a relation between closed geodesics for the theta group (the subgroup of the modular group generated by z+2 and -1 / z) and the continued fraction with even partial quotients. Using thermodynamic formalism, Tauberian results and the above-mentioned relation, we obtain the asymptotic growth number of closed trajectories for the theta group. Several results for the continued fraction expansion with even partial quotients are obtained; some of these are analogous to those already known for the usual continued fraction expansion related to the modular group, but our proofs are by necessity in general technically more difficult.Supported by The Netherlands Organization for Scientific Research (NWO).     

On Atkin and Swinnerton-Dyer congruence relations (3)

It is now well known that Hecke operators defined classically act trivially on genuine cuspforms for noncongruence subgroups of SL₂(Z). Atkin and Swinnerton-Dyer speculated the existence of p-adic Hecke operators so that the Fourier coefficients of their eigenfunctions satisfy three-term congruence recursions. In the previous two papers with the same title ([W.C. Li, L. Long, Z. Yang, On Atkin and Swinnerton-Dyer congruence relations, J. Number Theory 113 (1) (2005) 117-148] by W.C. Li, L. Long, Z. Yang and [A.O.L. Atkin, W.C. Li, L. Long, On Atkin and Swinnerton-Dyer congruence relations (2), Math. Ann. 340 (2) (2008) 335-358] by A.O.L. Atkin, W.C. Li, L. Long), the authors have studied two exceptional spaces of noncongruence cuspforms where almost all p-adic Hecke operators can be diagonalized simultaneously or semi-simultaneously. Moreover, it is shown that the l-adic Scholl representations attached to these spaces are modular in the sense that they are isomorphic, up to semisimplification, to the l-adic representations arising from classical automorphic forms.In this paper, we study an infinite family of spaces of noncongruence cuspforms (which includes the cases in [W.C. Li, L. Long, Z. Yang, On Atkin and Swinnerton-Dyer congruence relations, J. Number Theory 113 (1) (2005) 117-148; A.O.L. Atkin, W.C. Li, L. Long, On Atkin and Swinnerton-Dyer congruence relations (2), Math. Ann. 340 (2) (2008) 335-358]) under a general setting. It is shown that for each space in this family there exists a fixed basis so that the Fourier coefficients of each basis element satisfy certain weaker three-term congruence recursions. For a new case in this family, we will exhibit that the attached l-adic Scholl representations are modular and the p-adic Hecke operators can be diagonalized semi-simultaneously.     

On the lifting of elliptic cusp forms to cusp forms on quaternionic unitary groups

Let H be a definite quaternion algebra over Q with discriminant DH and R a maximal order of H. We denote by Gn a quaternionic unitary group and put Γn=Gn(Q)∩GL2n(R). Let Sκ(Γn) be the space of cusp forms of weight κ with respect to Γn on the quaternion half-space of degree n. We construct a lifting from primitive forms in Sk(SL₂(Z)) to Sk+2n−2(Γn) and a lifting from primitive forms in Sk(Γ₀(d)) to Sk+2(Γ₂), where d is a factor of DH. These liftings are generalizations of the Maass lifting investigated by Krieg.     

Algebraic independence of the values of Mahler functions associated with a certain continued fraction expansion

It is proved that the function , which can be expressed as a certain continued fraction, takes algebraically independent values at any distinct nonzero algebraic numbers inside the unit circle if the sequence {R_k}_k?0 is the generalized Fibonacci numbers.     

Superspecial abelian varieties and the Eichler Basis Problem for Hilbert modular forms

Let p be an unramified prime in a totally real field L such that h⁺(L)=1. Our main result shows that Hilbert modular newforms of parallel weight two for Γ₀(p) can be constructed naturally, via classical theta series, from modules of isogenies of superspecial abelian varieties with real multiplication on a Hilbert moduli space. This may be viewed as a geometric reinterpretation of the Eichler Basis Problem for Hilbert modular forms.     

Products of idempotent matrices over integral domains

We say that a ring R has the idempotent matrices property if every square singular matrix over R is a product of idempotent matrices. It is known that every field, and more generally, every Euclidean domain has the idempotent matrices property. In this paper we show that not every integral domain has the idempotent matrices property and that if a projective free ring has the idempotent matrices property then it must be a Bezout domain. We also show that a principal ideal domain has the idempotent matrices property if and only if every fraction a/b with b≠0 has a finite continued fraction expansion. New proofs are also provided for the results that every field and every Euclidean domain have the idempotent matrices property.     

On zeta function identities involving sums of squares and the zeta-theta correspondence

We prove two identities involving Dirichlet series, in the denominators of whose terms sums of two, three and four squares appear. These follow from two classical identities of Jacobi involving the four Jacobian Theta Functions θ₁(z;q), θ₂(z;q), θ₃(z;q) and θ₄(z;q), by the application of the Mellin transform. These results motivate the well-known correspondence between the set of the four Jacobian Theta Functions and the set of four classical zeta functions of which the Riemann Zeta Function is the third, and the Dirichlet Beta Function is the first.     

Some Theorems on the Rogers-Ramanujan Continued Fraction and Associated Theta Function Identities in Ramanujan's Lost Notebook

In his lost notebook, Ramanujan recorded several modular equations of degree 5 related to the Rogers-Ramanujan continued fraction R(q). We prove several of these identities and give factorizations of some of them in this paper.The parameter k = R(q) R₂(q₂) introduced by Ramanujan in his second notebook has not been recognized for its usefulness. In this work, we demonstrate how beautifully the parameter k works, as we prove several identities involving k stated by Ramanujan in the lost notebook.     

Twisted Hecke L-values and period polynomials

Let f₁,…,fd be an orthogonal basis for the space of cusp forms of even weight 2k on Γ₀(N). Let L(fi,s) and L(fi,χ,s) denote the L-function of fi and its twist by a Dirichlet character χ, respectively. In this note, we obtain a “trace formula” for the values at integers m and n with 0<m,n<2k and proper parity. In the case N=1 or N=2, the formula gives us a convenient way to evaluate precisely the value of the ratio L(f,χ,m)/L(f,n) for a Hecke eigenform f.     

Elliptic hypergeometric series on root systems

We derive a number of summation and transformation formulas for elliptic hypergeometric series on the root systems A_n, C_n and D_n. In the special cases of classical and q-series, our approach leads to new elementary proofs of the corresponding identities.