1. CJM 2017 (vol 70 pp. 481)
 Asakura, Masanori; Otsubo, Noriyuki

CM Periods, CM Regulators and Hypergeometric Functions, I
We prove the GrossDeligne conjecture on CM periods for motives
associated with $H^2$ of certain surfaces fibered over the projective
line. Then we prove for the same motives a formula which expresses
the $K_1$regulators in terms of hypergeometric functions ${}_3F_2$,
and obtain a new example of nontrivial regulators.
Keywords:period, regulator, complex multiplication, hypergeometric function Categories:14D07, 19F27, 33C20, 11G15, 14K22 

2. CJM 2016 (vol 69 pp. 687)
 Ovchinnikov, Alexey; Wibmer, Michael

Tannakian Categories with Semigroup Actions
Ostrowski's theorem implies that $\log(x),\log(x+1),\dots$ are
algebraically independent over $\mathbb{C}(x)$. More generally, for
a linear differential or difference equation, it is an important
problem to find all algebraic dependencies among a nonzero solution
$y$ and particular transformations of $y$, such as derivatives
of $y$ with respect to parameters, shifts of the arguments, rescaling,
etc. In the present paper, we develop a theory of Tannakian categories
with semigroup actions, which will be used to attack such questions
in full generality, as each linear differential equation gives
rise to a Tannakian category.
Deligne studied actions of braid groups on categories and obtained
a finite collection of axioms that characterizes such actions
to apply it to various geometric constructions. In this paper,
we find a finite set of axioms that characterizes actions of
semigroups that are finite free products of semigroups of the
form $\mathbb{N}^n\times
\mathbb{Z}/{n_1}\mathbb{Z}\times\cdots\times\mathbb{Z}/{n_r}\mathbb{Z}$
on Tannakian categories. This is the class of semigroups that
appear in many applications.
Keywords:semigroup actions on categories, Tannakian categories, difference algebraic groups, differential and difference equations with parameters Categories:18D10, 12H10, 20G05, 33C05, 33C80, 34K06 

3. 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, EisensteinKronecker series, $L$functions, hypergeometric series Categories:11F67, 33C20 

4. CJM 2012 (vol 64 pp. 721)
 Achab, Dehbia; Faraut, Jacques

Analysis of the BrylinskiKostant 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 KantorKoecherTits 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, KantorKoecherTits construction, Jordan algebra, Bernstein identity, Meijer $G$function Categories:17C36, 22E46, 32M15, 33C80 

5. 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 measure Categories:60G50, 33C20, 34M25, 44A10 

6. 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 HallLittlewood
polynomial. Our conjecture refines several earlier conjectures concerning
the space of diagonal harmonics including the ``shuffle conjecture"
(Duke J. Math. $\mathbf {126}$ ($2005$), 195232) for $\nabla e_n[X]$.
We bring to light that certain generalized HallLittlewood 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, HallLittlewood symmetric functions Categories:05E05, 33D52 

7. 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
AppellLerch 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, AppellLerch sum, generalized Lambert series Categories:11B65, 33D15 

8. 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 verywellpoised basic hypergeometric series, orthonormal polynomials in one and two variables, Racah and $q$Racah polynomials and their extensions Categories:33D45, 33D50 

9. 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 AskeyWilson 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, AskeyWilson operator, $q$difference operator, recursion coefficients Categories:33D45, 42C05 

10. 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
$$
\F\_{\lambda,p}=\sup_{0\leq r
< 1}(1r^{2})^{\mathcal{R}e[i\lambda]\frac{n}{2}+l}\Big[\int_{S}F(ru)^{p}du
\Big]^{\frac{1}{p}} < +\infty.
$$
Keywords:Lie ball, Shilov boundary, Fatou's theorem, hyperfuctions, parabolic subgroup, homogeneous line bundle Categories:32A45, 30E20, 33C67, 33C60, 33C55, 32A25, 33C75, 11K70 

11. CJM 2010 (vol 62 pp. 261)
12. 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 

13. CJM 2006 (vol 58 pp. 726)
 Chiang, YikMan; 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 zerodistribution 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
(finitezeros) 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 Proble Categories:34M10, 33C15, 33C47 

14. 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 HodgeTate 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 HodgeTate structures and make a conjecture relating the variations
of mixed HodgeTate structures of multiple logarithms to those of
general multiple {\em poly}\/logarithms. Then following
Deligne and Beilinson we describe an
approach to defining the singlevalued
real analytic version of the multiple polylogarithms which
generalizes the wellknown result of Zagier on
classical polylogarithms. In the process we find some interesting
identities relating singlevalued 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 singlevalued
version of multiple polylogarithms.
Categories:14D07, 14D05, 33B30 

15. CJM 2004 (vol 56 pp. 897)
 Borwein, Jonathan M.; Borwein, David; Galway, William F.

Finding and Excluding $b$ary MachinType 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{Machintype BBP formulae}, for which it
is relatively easy to determine whether or not a given constant
$\kappa$ has a Machintype BBP formula. In particular, given $b \in
\mathbb{N}$, $b>2$, $b$ not a proper power, a $b$ary Machintype
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 $a_m \arctan(1/(12^m))$. Of
particular interest, we show that $\pi$ has no Machintype BBP
arctangent formula when $b \neq 2$. To the best of our knowledge,
when there is no Machintype BBP formula for a constant then no BBP
formula of any form is known for that constant.
Keywords:BBP formulae, Machintype formulae, arctangents,, logarithms, normality, Mersenne primes, Bang's theorem,, Zsigmondy's theorem, primitive prime factors, $p$adic analysis Categories:11Y99, 11A51, 11Y50, 11K36, 33B10 

16. CJM 2003 (vol 55 pp. 1134)
 Casarino, Valentina

Norms of Complex Harmonic Projection Operators
In this paper we estimate the $(L^pL^2)$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 

17. 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, AlSalamChihara polynomials, continuous $q$ultraspherical polynomials Categories:33D45, 33D20, 33C45, 30E05 

18. 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 

19. 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 $\int_0^1
u^{\alpha} (1u)^{\alpha} f(u) \,du$ 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 $z = y/(y1)$ 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 

20. CJM 2000 (vol 52 pp. 31)
 Chan, Heng Huat; Liaw, WenChin

On RussellType Modular Equations
In this paper, we revisit Russelltype 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 Russelltype modular equations in the theory
of signature~$3$.
Categories:33D10, 33C05, 11F11 

21. 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 probabilitypreserving, but lead to socalled 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 functions Categories:43A62, 33C25, 43A20, 43A90 

22. CJM 1998 (vol 50 pp. 1236)
23. 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 

24. 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 rankone
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 (multivalued) 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, hyperlogarithms Categories:35C05, 33E30, 33C45, 34B27, 35J40 

25. 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 group Categories:33C50, 33C45 
