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1. CJM Online first
Mahler Measures as Linear Combinations of $L$-values of Multiple Modular Forms We study the Mahler measures of certain families of Laurent
polynomials in two and three variables. Each of the known Mahler
measure formulas for these families involves $L$-values of at most one
newform and/or at most one quadratic character. In this paper, we
show, either rigorously or numerically, that the Mahler measures of
some polynomials are related to $L$-values of multiple newforms and
quadratic characters simultaneously. The results suggest that the
number of modular $L$-values appearing in the formulas significantly
depends on the shape of the algebraic value of the parameter chosen
for each polynomial. As a consequence, we also obtain new formulas
relating special values of hypergeometric series evaluated at
algebraic numbers to special values of $L$-functions.
Keywords:Mahler measures, Eisenstein-Kronecker series, $L$-functions, hypergeometric series Categories:11F67, 33C20 |
2. CJM 2014 (vol 66 pp. 1078)
Values of Twisted Tensor $L$-functions of Automorphic Forms Over Imaginary Quadratic Fields Let $K$ be a complex quadratic extension of $\mathbb{Q}$ and let $\mathbb{A}_K$
denote the adeles of $K$.
We find special values at all of the critical points of twisted
tensor $L$-functions attached to cohomological cuspforms on $GL_2(\mathbb{A}_K)$,
and establish Galois equivariance of the values.
To investigate the values, we determine the archimedean factors
of a class of integral representations of these $L$-functions,
thus proving a conjecture due to Ghate. We also investigate
analytic properties of these $L$-functions, such as their functional
equations.
Keywords:twisted tensor $L$-function, cuspform, hypergeometric series Categories:11F67, 11F37 |
3. CJM 2010 (vol 63 pp. 200)
An Explicit Polynomial Expression for a $q$-Analogue of the 9-$j$ Symbols Using standard transformation and summation formulas for basic
hypergeometric series we obtain an explicit polynomial form of the
$q$-analogue of the 9-$j$ symbols, introduced by the author in a
recent publication. We also consider a limiting case in which the
9-$j$ symbol factors into two Hahn polynomials. The same
factorization occurs in another limit case of the corresponding
$q$-analogue.
Keywords:6-$j$ and 9-$j$ symbols, $q$-analogues, balanced and very-well-poised basic hypergeometric series, orthonormal polynomials in one and two variables, Racah and $q$-Racah polynomials and their extensions Categories:33D45, 33D50 |
4. CJM 2000 (vol 52 pp. 961)
Algebraic Evaluations of Some Euler Integrals, Duplication Formulae for Appell's Hypergeometric Function $F_1$, and Brownian Variations |
Algebraic Evaluations of Some Euler Integrals, Duplication Formulae for Appell's Hypergeometric Function $F_1$, and Brownian Variations Explicit evaluations of the symmetric Euler integral $\int_0^1
u^{\alpha} (1-u)^{\alpha} f(u) \,du$ are obtained for some particular
functions $f$. These evaluations are related to duplication formulae
for Appell's hypergeometric function $F_1$ which give reductions of
$F_1 (\alpha, \beta, \beta, 2 \alpha, y, z)$ in terms of more
elementary functions for arbitrary $\beta$ with $z = y/(y-1)$ and for
$\beta = \alpha + \half$ with arbitrary $y$, $z$. These duplication
formulae generalize the evaluations of some symmetric Euler integrals
implied by the following result: if a standard Brownian bridge is
sampled at time $0$, time $1$, and at $n$ independent random times
with uniform distribution on $[0,1]$, then the broken line
approximation to the bridge obtained from these $n+2$ values has a
total variation whose mean square is $n(n+1)/(2n+1)$.
Keywords:Brownian bridge, Gauss's hypergeometric function, Lauricella's multiple hypergeometric series, uniform order statistics, Appell functions Categories:33C65, 60J65 |
5. CJM 1997 (vol 49 pp. 543)
Some summation theorems and transformations for $q$-series We introduce a double sum extension of a very well-poised series and
extend to this the transformations of Bailey and Sears as well as the
${}_6\f_5$ summation formula of F.~H.~Jackson and the $q$-Dixon sum.
We also give $q$-integral representations of the double sum.
Generalizations of the Nassrallah-Rahman integral are also found.
Keywords:Basic hypergeometric series, balanced series,, very well-poised series, integral representations,, Al-Salam-Chihara polynomials. Categories:33D20, 33D60 |