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1. CJM 2014 (vol 67 pp. 424)

Samart, Detchat
 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 seriesCategories:11F67, 33C20

2. CJM 2012 (vol 64 pp. 721)

Achab, Dehbia; Faraut, Jacques
 Analysis of the Brylinski-Kostant Model for Spherical Minimal Representations We revisit with another view point the construction by R. Brylinski and B. Kostant of minimal representations of simple Lie groups. We start from a pair $(V,Q)$, where $V$ is a complex vector space and $Q$ a homogeneous polynomial of degree 4 on $V$. The manifold $\Xi$ is an orbit of a covering of ${\rm Conf}(V,Q)$, the conformal group of the pair $(V,Q)$, in a finite dimensional representation space. By a generalized Kantor-Koecher-Tits construction we obtain a complex simple Lie algebra $\mathfrak g$, and furthermore a real form ${\mathfrak g}_{\mathbb R}$. The connected and simply connected Lie group $G_{\mathbb R}$ with ${\rm Lie}(G_{\mathbb R})={\mathfrak g}_{\mathbb R}$ acts unitarily on a Hilbert space of holomorphic functions defined on the manifold $\Xi$. Keywords:minimal representation, Kantor-Koecher-Tits construction, Jordan algebra, Bernstein identity, Meijer $G$-functionCategories:17C36, 22E46, 32M15, 33C80

3. CJM 2011 (vol 64 pp. 961)

Borwein, Jonathan M.; Straub, Armin; Wan, James; Zudilin, Wadim
 Densities of Short Uniform Random Walks We study the densities of uniform random walks in the plane. A special focus is on the case of short walks with three or four steps and less completely those with five steps. As one of the main results, we obtain a hypergeometric representation of the density for four steps, which complements the classical elliptic representation in the case of three steps. It appears unrealistic to expect similar results for more than five steps. New results are also presented concerning the moments of uniform random walks and, in particular, their derivatives. Relations with Mahler measures are discussed. Keywords:random walks, hypergeometric functions, Mahler measureCategories:60G50, 33C20, 34M25, 44A10

4. CJM 2011 (vol 64 pp. 822)

Haglund, J.; Morse, J.; Zabrocki, M.
 A Compositional Shuffle Conjecture Specifying Touch Points of the Dyck Path We introduce a $q,t$-enumeration of Dyck paths that are forced to touch the main diagonal at specific points and forbidden to touch elsewhere and conjecture that it describes the action of the Macdonald theory $\nabla$ operator applied to a Hall--Littlewood polynomial. Our conjecture refines several earlier conjectures concerning the space of diagonal harmonics including the shuffle conjecture" (Duke J. Math. $\mathbf {126}$ ($2005$), 195-232) for $\nabla e_n[X]$. We bring to light that certain generalized Hall--Littlewood polynomials indexed by compositions are the building blocks for the algebraic combinatorial theory of $q,t$-Catalan sequences, and we prove a number of identities involving these functions. Keywords:Dyck Paths, Parking functions, Hall--Littlewood symmetric functionsCategories:05E05, 33D52

5. CJM 2011 (vol 64 pp. 935)

McIntosh, Richard J.
 The H and K Families of Mock Theta Functions In his last letter to Hardy, Ramanujan defined 17 functions $F(q)$, $|q|\lt 1$, which he called mock $\theta$-functions. He observed that as $q$ radially approaches any root of unity $\zeta$ at which $F(q)$ has an exponential singularity, there is a $\theta$-function $T_\zeta(q)$ with $F(q)-T_\zeta(q)=O(1)$. Since then, other functions have been found that possess this property. These functions are related to a function $H(x,q)$, where $x$ is usually $q^r$ or $e^{2\pi i r}$ for some rational number $r$. For this reason we refer to $H$ as a universal'' mock $\theta$-function. Modular transformations of $H$ give rise to the functions $K$, $K_1$, $K_2$. The functions $K$ and $K_1$ appear in Ramanujan's lost notebook. We prove various linear relations between these functions using Appell-Lerch sums (also called generalized Lambert series). Some relations (mock theta conjectures'') involving mock $\theta$-functions of even order and $H$ are listed. Keywords:mock theta function, $q$-series, Appell-Lerch sum, generalized Lambert seriesCategories:11B65, 33D15

6. CJM 2010 (vol 63 pp. 200)

Rahman, Mizan
 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 extensionsCategories:33D45, 33D50

7. CJM 2010 (vol 63 pp. 181)

Ismail, Mourad E. H.; Obermaier, Josef
 Characterizations of Continuous and Discrete $q$-Ultraspherical Polynomials We characterize the continuous $q$-ultraspherical polynomials in terms of the special form of the coefficients in the expansion $\mathcal{D}_q P_n(x)$ in the basis $\{P_n(x)\}$, $\mathcal{D}_q$ being the Askey--Wilson divided difference operator. The polynomials are assumed to be symmetric, and the connection coefficients are multiples of the reciprocal of the square of the $L^2$ norm of the polynomials. A similar characterization is given for the discrete $q$-ultraspherical polynomials. A new proof of the evaluation of the connection coefficients for big $q$-Jacobi polynomials is given. Keywords:continuous $q$-ultraspherical polynomials, big $q$-Jacobi polynomials, discrete $q$-ultra\-spherical polynomials, Askey--Wilson operator, $q$-difference operator, recursion coefficientsCategories:33D45, 42C05

8. CJM 2010 (vol 62 pp. 1276)

El Wassouli, Fouzia
 A Generalized Poisson Transform of an $L^{p}$-Function over the Shilov Boundary of the $n$-Dimensional Lie Ball Let $\mathcal{D}$ be the $n$-dimensional Lie ball and let $\mathbf{B}(S)$ be the space of hyperfunctions on the Shilov boundary $S$ of $\mathcal{D}$. The aim of this paper is to give a necessary and sufficient condition on the generalized Poisson transform $P_{l,\lambda}f$ of an element $f$ in the space $\mathbf{B}(S)$ for $f$ to be in $L^{p}(S)$, $1 > p > \infty.$ Namely, if $F$ is the Poisson transform of some $f\in \mathbf{B}(S)$ (i.e., $F=P_{l,\lambda}f$), then for any $l\in \mathbb{Z}$ and $\lambda\in \mathbb{C}$ such that $\mathcal{R}e[i\lambda] > \frac{n}{2}-1$, we show that $f\in L^{p}(S)$ if and only if $f$ satisfies the growth condition $$\|F\|_{\lambda,p}=\sup_{0\leq r < 1}(1-r^{2})^{\mathcal{R}e[i\lambda]-\frac{n}{2}+l}\Big[\int_{S}|F(ru)|^{p}du \Big]^{\frac{1}{p}} < +\infty.$$ Keywords:Lie ball, Shilov boundary, Fatou's theorem, hyperfuctions, parabolic subgroup, homogeneous line bundleCategories:32A45, 30E20, 33C67, 33C60, 33C55, 32A25, 33C75, 11K70

9. CJM 2010 (vol 62 pp. 261)

Chiang, Yik-Man; Ismail, Mourad E. H.
 Erratum to: On Value Distribution Theory of Second Order Periodic ODEs, Special Functions and Orthogonal Polynomials No abstract. Keywords:Complex Oscillation theory, Exponent of convergence of zeros, zero distribution of Bessel and Confluent hypergeometric functions, Lommel transform, Bessel polynomials, Heine ProblemCategories:34M10, 33C15, 33C47

10. CJM 2009 (vol 61 pp. 373)

McKee, Mark
 An Infinite Order Whittaker Function In this paper we construct a flat smooth section of an induced space $I(s,\eta)$ of $SL_2(\mathbb{R})$ so that the attached Whittaker function is not of finite order. An asymptotic method of classical analysis is used. Categories:11F70, 22E45, 41A60, 11M99, 30D15, 33C15

11. CJM 2006 (vol 58 pp. 726)

Chiang, Yik-Man; Ismail, Mourad E. H.
 On Value Distribution Theory of Second Order Periodic ODEs, Special Functions and Orthogonal Polynomials We show that the value distribution (complex oscillation) of solutions of certain periodic second order ordinary differential equations studied by Bank, Laine and Langley is closely related to confluent hypergeometric functions, Bessel functions and Bessel polynomials. As a result, we give a complete characterization of the zero-distribution in the sense of Nevanlinna theory of the solutions for two classes of the ODEs. Our approach uses special functions and their asymptotics. New results concerning finiteness of the number of zeros (finite-zeros) problem of Bessel and Coulomb wave functions with respect to the parameters are also obtained as a consequence. We demonstrate that the problem for the remaining class of ODEs not covered by the above special function approach" can be described by a classical Heine problem for differential equations that admit polynomial solutions. Keywords:Complex Oscillation theory, Exponent of convergence of zeros, zero distribution of Bessel and Confluent hypergeometric functions, Lommel transform, Bessel polynomials, Heine ProbleCategories:34M10, 33C15, 33C47

12. CJM 2004 (vol 56 pp. 1308)

Zhao, Jianqiang
 Variations of Mixed Hodge Structures of Multiple Polylogarithms It is well known that multiple polylogarithms give rise to good unipotent variations of mixed Hodge-Tate structures. In this paper we shall {\em explicitly} determine these structures related to multiple logarithms and some other multiple polylogarithms of lower weights. The purpose of this explicit construction is to give some important applications: First we study the limit of mixed Hodge-Tate structures and make a conjecture relating the variations of mixed Hodge-Tate structures of multiple logarithms to those of general multiple {\em poly}\/logarithms. Then following Deligne and Beilinson we describe an approach to defining the single-valued real analytic version of the multiple polylogarithms which generalizes the well-known result of Zagier on classical polylogarithms. In the process we find some interesting identities relating single-valued multiple polylogarithms of the same weight $k$ when $k=2$ and 3. At the end of this paper, motivated by Zagier's conjecture we pose a problem which relates the special values of multiple Dedekind zeta functions of a number field to the single-valued version of multiple polylogarithms. Categories:14D07, 14D05, 33B30

13. CJM 2004 (vol 56 pp. 897)

Borwein, Jonathan M.; Borwein, David; Galway, William F.
 Finding and Excluding $b$-ary Machin-Type Individual Digit Formulae Constants with formulae of the form treated by D.~Bailey, P.~Borwein, and S.~Plouffe (\emph{BBP formulae} to a given base $b$) have interesting computational properties, such as allowing single digits in their base $b$ expansion to be independently computed, and there are hints that they should be \emph{normal} numbers, {\em i.e.,} that their base $b$ digits are randomly distributed. We study a formally limited subset of BBP formulae, which we call \emph{Machin-type BBP formulae}, for which it is relatively easy to determine whether or not a given constant $\kappa$ has a Machin-type BBP formula. In particular, given $b \in \mathbb{N}$, $b>2$, $b$ not a proper power, a $b$-ary Machin-type BBP arctangent formula for $\kappa$ is a formula of the form $\kappa = \sum_{m} a_m \arctan(-b^{-m})$, $a_m \in \mathbb{Q}$, while when $b=2$, we also allow terms of the form $a_m \arctan(1/(1-2^m))$. Of particular interest, we show that $\pi$ has no Machin-type BBP arctangent formula when $b \neq 2$. To the best of our knowledge, when there is no Machin-type BBP formula for a constant then no BBP formula of any form is known for that constant. Keywords:BBP formulae, Machin-type formulae, arctangents,, logarithms, normality, Mersenne primes, Bang's theorem,, Zsigmondy's theorem, primitive prime factors, $p$-adic analysisCategories:11Y99, 11A51, 11Y50, 11K36, 33B10

14. CJM 2003 (vol 55 pp. 1134)

Casarino, Valentina
 Norms of Complex Harmonic Projection Operators In this paper we estimate the $(L^p-L^2)$-norm of the complex harmonic projectors $\pi_{\ell\ell'}$, $1\le p\le 2$, uniformly with respect to the indexes $\ell,\ell'$. We provide sharp estimates both for the projectors $\pi_{\ell\ell'}$, when $\ell,\ell'$ belong to a proper angular sector in $\mathbb{N} \times \mathbb{N}$, and for the projectors $\pi_{\ell 0}$ and $\pi_{0 \ell}$. The proof is based on an extension of a complex interpolation argument by C.~Sogge. In the appendix, we prove in a direct way the uniform boundedness of a particular zonal kernel in the $L^1$ norm on the unit sphere of $\mathbb{R}^{2n}$. Categories:43A85, 33C55, 42B15

15. CJM 2002 (vol 54 pp. 709)

Ismail, Mourad E. H.; Stanton, Dennis
 $q$-Integral and Moment Representations for $q$-Orthogonal Polynomials We develop a method for deriving integral representations of certain orthogonal polynomials as moments. These moment representations are applied to find linear and multilinear generating functions for $q$-orthogonal polynomials. As a byproduct we establish new transformation formulas for combinations of basic hypergeometric functions, including a new representation of the $q$-exponential function $\mathcal{E}_q$. Keywords:$q$-integral, $q$-orthogonal polynomials, associated polynomials, $q$-difference equations, generating functions, Al-Salam-Chihara polynomials, continuous $q$-ultraspherical polynomialsCategories:33D45, 33D20, 33C45, 30E05

16. CJM 2002 (vol 54 pp. 239)

Cartwright, Donald I.; Steger, Tim
 Elementary Symmetric Polynomials in Numbers of Modulus $1$ We describe the set of numbers $\sigma_k(z_1,\ldots,z_{n+1})$, where $z_1,\ldots,z_{n+1}$ are complex numbers of modulus $1$ for which $z_1z_2\cdots z_{n+1}=1$, and $\sigma_k$ denotes the $k$-th elementary symmetric polynomial. Consequently, we give sharp constraints on the coefficients of a complex polynomial all of whose roots are of the same modulus. Another application is the calculation of the spectrum of certain adjacency operators arising naturally on a building of type ${\tilde A}_n$. Categories:05E05, 33C45, 30C15, 51E24

17. CJM 2000 (vol 52 pp. 961)

Ismail, Mourad E. H.; Pitman, Jim
 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 functionsCategories:33C65, 60J65

18. CJM 2000 (vol 52 pp. 31)

Chan, Heng Huat; Liaw, Wen-Chin
 On Russell-Type Modular Equations In this paper, we revisit Russell-type modular equations, a collection of modular equations first studied systematically by R.~Russell in 1887. We give a proof of Russell's main theorem and indicate the relations between such equations and the constructions of Hilbert class fields of imaginary quadratic fields. Motivated by Russell's theorem, we state and prove its cubic analogue which allows us to construct Russell-type modular equations in the theory of signature~$3$. Categories:33D10, 33C05, 11F11

19. CJM 1999 (vol 51 pp. 96)

Rösler, Margit; Voit, Michael
 Partial Characters and Signed Quotient Hypergroups If $G$ is a closed subgroup of a commutative hypergroup $K$, then the coset space $K/G$ carries a quotient hypergroup structure. In this paper, we study related convolution structures on $K/G$ coming from deformations of the quotient hypergroup structure by certain functions on $K$ which we call partial characters with respect to $G$. They are usually not probability-preserving, but lead to so-called signed hypergroups on $K/G$. A first example is provided by the Laguerre convolution on $\left[ 0,\infty \right[$, which is interpreted as a signed quotient hypergroup convolution derived from the Heisenberg group. Moreover, signed hypergroups associated with the Gelfand pair $\bigl( U(n,1), U(n) \bigr)$ are discussed. Keywords:quotient hypergroups, signed hypergroups, Laguerre convolution, Jacobi functionsCategories:43A62, 33C25, 43A20, 43A90

20. CJM 1998 (vol 50 pp. 1236)

Kalton, N. J.; Tzafriri, L.
 The behaviour of Legendre and ultraspherical polynomials in $L_p$-spaces We consider the analogue of the $\Lambda(p)-$problem for subsets of the Legendre polynomials or more general ultraspherical polynomials. We obtain the best possible'' result that if $2 Categories:42C10, 33C45, 46B07 21. CJM 1998 (vol 50 pp. 412) McIntosh, Richard J.  Asymptotic transformations of$q$-series For the$q$-series$\sum_{n=0}^\infty a^nq^{bn^2+cn}/(q)_n$we construct a companion$q$-series such that the asymptotic expansions of their logarithms as$q\to 1^{\scriptscriptstyle -}$differ only in the dominant few terms. The asymptotic expansion of their quotient then has a simple closed form; this gives rise to a new$q$-hypergeometric identity. We give an asymptotic expansion of a general class of$q$-series containing some of Ramanujan's mock theta functions and Selberg's identities. Categories:11B65, 33D10, 34E05, 41A60 22. CJM 1998 (vol 50 pp. 40) Engliš, Miroslav; Peetre, Jaak  Green's functions for powers of the invariant Laplacian The aim of the present paper is the computation of Green's functions for the powers$\DDelta^m$of the invariant Laplace operator on rank-one Hermitian symmetric spaces. Starting with the noncompact case, the unit ball in$\CC^d$, we obtain a complete result for$m=1,2$in all dimensions. For$m\ge3$the formulas grow quite complicated so we restrict ourselves to the case of the unit disc ($d=1$) where we develop a method, possibly applicable also in other situations, for reducing the number of integrations by half, and use it to give a description of the boundary behaviour of these Green functions and to obtain their (multi-valued) analytic continuation to the entire complex plane. Next we discuss the type of special functions that turn up (hyperlogarithms of Kummer). Finally we treat also the compact case of the complex projective space$\Bbb P^d$(for$d=1$, the Riemann sphere) and, as an application of our results, use eigenfunction expansions to obtain some new identities involving sums of Legendre ($d=1$) or Jacobi ($d>1$) polynomials and the polylogarithm function. The case of Green's functions of powers of weighted (no longer invariant, but only covariant) Laplacians is also briefly discussed. Keywords:Invariant Laplacian, Green's functions, dilogarithm, trilogarithm, Legendre and Jacobi polynomials, hyperlogarithmsCategories:35C05, 33E30, 33C45, 34B27, 35J40 23. CJM 1998 (vol 50 pp. 193) Xu, Yuan  Intertwining operator and$h$-harmonics associated with reflection groups We study the intertwining operator and$h$-harmonics in Dunkl's theory on$h$-harmonics associated with reflection groups. Based on a biorthogonality between the ordinary harmonics and the action of the intertwining operator$V$on the harmonics, the main result provides a method to compute the action of the intertwining operator$V$on polynomials and to construct an orthonormal basis for the space of$h$-harmonics. Keywords:$h$-harmonics, intertwining operator, reflection groupCategories:33C50, 33C45 24. CJM 1997 (vol 49 pp. 1224) Ørsted, Bent; Zhang, Genkai  Tensor products of analytic continuations of holomorphic discrete series We give the irreducible decomposition of the tensor product of an analytic continuation of the holomorphic discrete series of$\SU(2, 2)$with its conjugate. Keywords:Holomorphic discrete series, tensor product, spherical function, Clebsch-Gordan coefficient, Plancherel theoremCategories:22E45, 43A85, 32M15, 33A65 25. CJM 1997 (vol 49 pp. 543) Ismail, Mourad E. H.; Rahman, Mizan; Suslov, Sergei K.  Some summation theorems and transformations for$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
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