1. CMB 2017 (vol 61 pp. 473)
 Awonusika, Richard; Taheri, Ali

A spectral identity on Jacobi polynomials and its analytic implications
The Jacobi coefficients $c^{\ell}_{j}(\alpha,\beta)$ ($1\leq
j\leq \ell$, $\alpha,\beta\gt 1$) are linked to the Maclaurin
spectral expansion of the Schwartz kernel of functions of the
Laplacian on a compact rank one symmetric space. It
is proved that these coefficients can be computed by transforming
the even derivatives of the the Jacobi polynomials $P_{k}^{(\alpha,\beta)}$ ($k\geq 0, \alpha,\beta\gt 1$) into a spectral sum associated with
the Jacobi operator. The first few coefficients are explicitly
computed and a direct trace
interpretation of the Maclaurin coefficients is presented.
Keywords:Jacobi coefficient, LaplaceBeltrami operator, symmetric space, Maclaurin expansion, Jacobi polynomial Categories:33C05, 33C45, 35A08, 35C05, 35C10, 35C15 

2. CMB 2015 (vol 58 pp. 877)
 Zaatra, Mohamed

Generating Some Symmetric Semiclassical Orthogonal Polynomials
We show that if $v$ is a regular semiclassical form
(linear functional), then the symmetric form $u$ defined by the
relation
$x^{2}\sigma u = \lambda v$,
where $(\sigma f)(x)=f(x^{2})$ and the odd
moments of $u$ are $0$, is also
regular and semiclassical form for every
complex $\lambda $ except for a discrete set of numbers depending
on $v$. We give explicitly the threeterm recurrence relation
and the
structure relation coefficients of the orthogonal polynomials
sequence associated with $u$ and the class of the form $u$ knowing
that of $v$. We conclude with an illustrative example.
Keywords:orthogonal polynomials, quadratic decomposition, semiclassical forms, structure relation Categories:33C45, 42C05 

3. CMB 2015 (vol 59 pp. 136)
 Kajihara, Yasushi

Transformation Formulas for Bilinear Sums of Basic Hypergeometric Series
A master formula of transformation formulas for bilinear sums
of basic hypergeometric series
is proposed.
It is obtained from the author's previous results on
a transformation formula for Milne's multivariate generalization
of basic hypergeometric
series of type $A$ with different dimensions and it can be considered
as a
generalization of the WhippleSears transformation formula for
terminating balanced ${}_4 \phi_3$
series.
As an application of the master formula, the one variable cases
of some transformation formulas
for bilinear sums of basic hypergeometric series are given as
examples.
The bilinear transformation formulas seem to be new in the literature,
even in one variable case.
Keywords:bilinear sums, basic hypergeometric series Category:33D20 

4. CMB 2015 (vol 58 pp. 423)
 Yamagishi, Masakazu

Resultants of Chebyshev Polynomials: The First, Second, Third, and Fourth Kinds
We give an explicit formula for the resultant of Chebyshev polynomials of the
first, second, third, and fourth kinds.
We also compute the resultant of modified cyclotomic polynomials.
Keywords:resultant, Chebyshev polynomial, cyclotomic polynomial Categories:11R09, 11R18, 12E10, 33C45 

5. CMB 2011 (vol 55 pp. 571)
 Miller, A. R.; Paris, R. B.

A Generalised KummerType Transformation for the ${}_pF_p(x)$ Hypergeometric Function
In a recent paper, Miller derived a Kummertype
transformation for the generalised hypergeometric function ${}_pF_p(x)$ when pairs of
parameters differ by unity, by means of a reduction
formula for a certain KampÃ© de FÃ©riet function. An alternative and simpler derivation of this
transformation is obtained here by application of the wellknown Kummer transformation for the
confluent hypergeometric function corresponding to $p=1$.
Keywords:generalised hypergeometric series, Kummer transformation Categories:33C15, 33C20 

6. CMB 2010 (vol 54 pp. 538)
7. CMB 2009 (vol 52 pp. 583)
8. CMB 2009 (vol 40 pp. 276)
 Chouikha, Raouf

Fonctions elliptiques et Ã©quations diffÃ©rentielles ordinaires
In this paper, we detail some results of a previous note concerning
a trigonometric expansion of the Weierstrass elliptic function
$\{\wp(z);\, 2\omega, 2\omega'\}$. In particular, this implies its
classical Fourier expansion. We use a direct integration method of
the ODE $$(E)\left\{\matrix{{d^2u \over dt^2} = P(u, \lambda)\hfill \cr
u(0) = \sigma\hfill \cr {du \over dt}(0) = \tau\hfill \cr}\right.$$
where $P(u)$ is a polynomial of degree $n = 2$ or $3$. In this case,
the bifurcations of $(E)$ depend on one parameter only. Moreover, this
global method seems not to apply to the cases $n > 3$.
Categories:33E05, 34A05, 33E20, 33E30, 34A20, 34C23 

9. CMB 2008 (vol 51 pp. 561)
10. CMB 2008 (vol 51 pp. 627)
 Vidanovi\'{c}, Mirjana V.; Tri\v{c}kovi\'{c}, Slobodan B.; Stankovi\'{c}, Miomir S.

Summation of Series over Bourget Functions
In this paper we derive formulas for summation of series involving
J.~Bourget's generalization of Bessel functions of integer order, as
well as the analogous generalizations by H.~M.~Srivastava. These series are
expressed in terms of the Riemann $\z$ function and Dirichlet
functions $\eta$, $\la$, $\b$, and can be brought into closed form in
certain cases, which means that the infinite series are represented
by finite sums.
Keywords:Riemann zeta function, Bessel functions, Bourget functions, Dirichlet functions Categories:33C10, 11M06, 65B10 

11. CMB 2007 (vol 50 pp. 547)
 Iakovlev, Serguei

Inverse Laplace Transforms Encountered in Hyperbolic Problems of NonStationary FluidStructure Interaction
The paper offers a study of the inverse Laplace
transforms of the functions $I_n(rs)\{sI_n^{'}(s)\}^{1}$ where
$I_n$ is the modified Bessel function of the first kind and $r$ is
a parameter. The present study is a continuation of the author's
previous work %[\textit{Canadian Mathematical Bulletin} 45]
on the
singular behavior of the special case of the functions in
question, $r$=1. The general case of $r \in [0,1]$ is addressed,
and it is shown that the inverse Laplace transforms for such $r$
exhibit significantly more complex behavior than their
predecessors, even though they still only have two different types
of points of discontinuity: singularities and finite
discontinuities. The functions studied originate from
nonstationary fluidstructure interaction, and as such are of
interest to researchers working in the area.
Categories:44A10, 44A20, 33C10, 40A30, 74F10, 76Q05 

12. CMB 2007 (vol 50 pp. 284)
 McIntosh, Richard J.

Second Order Mock Theta Functions
In his last letter to Hardy, Ramanujan
defined 17 functions $F(q)$, where $q<1$. He called them mock theta
functions, because as $q$ radially approaches any point $e^{2\pi ir}$
($r$ rational), there is a theta function $F_r(q)$ with $F(q)F_r(q)=O(1)$.
In this paper we establish the relationship between two families of mock
theta functions.
Keywords:$q$series, mock theta function, Mordell integral Categories:11B65, 33D15 

13. CMB 2005 (vol 48 pp. 382)
 De Carli, Laura

Uniform Estimates of Ultraspherical Polynomials of Large Order
In this paper we prove the sharp inequality
$$ P_n^{(s)}(x)\leq
P_n^{(s)}(1)\bigl(x^n +\frac{n1}{2 s+1}(1x^n)\bigr),$$
where
$P_n^{(s)}(x)$ is the classical ultraspherical polynomial of
degree $n$ and order $s\ge n\frac{1+\sqrt 5}{4}$. This inequality
can be refined in $[0,z_n^s]$ and $[z_n^s,1]$, where $z_n^s$
denotes the largest zero of $P_n^{(s)}(x)$.
Categories:42C05, 33C47 

14. CMB 2005 (vol 48 pp. 147)
 Väänänen, Keijo; Zudilin, Wadim

BakerType Estimates for Linear Forms in the Values of $q$Series
We obtain lower estimates for the absolute values
of linear forms of the values of generalized Heine
series at nonzero points of an imaginary quadratic field~$\II$,
in particular of the values of $q$exponential function.
These estimates depend on the individual coefficients,
not only on the maximum of their absolute values.
The proof uses a variant of classical Siegel's method
applied to a system of functional Poincar\'etype equations
and the connection between the solutions of these functional
equations and the generalized Heine series.
Keywords:measure of linear independence, $q$series Categories:11J82, 33D15 

15. CMB 2002 (vol 45 pp. 567)
 De Sole, Alberto; Kac, Victor G.

Subalgebras of $\gc_N$ and Jacobi Polynomials
We classify the subalgebras of the general Lie conformal algebra
$\gc_N$ that act irreducibly on $\mathbb{C} [\partial]^N$ and that
are normalized by the sl$_2$part of a Virasoro element. The
problem turns out to be closely related to classical Jacobi
polynomials $P_n^{(\sigma,\sigma)}$, $\sigma \in \mathbb{C}$. The
connection goes both wayswe use in our classification some
classical properties of Jacobi polynomials, and we derive from the
theory of conformal algebras some apparently new properties of
Jacobi polynomials.
Categories:17B65, 17B68, 17B69, 33C45 

16. CMB 2002 (vol 45 pp. 436)
17. CMB 2002 (vol 45 pp. 257)
 Lee, Min Ho

Modular Forms Associated to Theta Functions
We use the theory of Jacobilike forms to construct modular forms for a
congruence subgroup of $\SL(2,\mathbb{R})$ which can be expressed as linear
combinations of products of certain theta functions.
Categories:11F11, 11F27, 33D10 

18. CMB 2001 (vol 44 pp. 337)
 Vinet, Luc; Zhedanov, Alexei

Spectral Transformations of the Laurent Biorthogonal Polynomials, II. Pastro Polynomials
We continue to study the simplest closure conditions for chains of
spectral transformations of the Laurent biorthogonal polynomials
($\LBP$). It is shown that the 11periodic $q$closure condition
leads to the $\LBP$ introduced by Pastro. We introduce classes of
semiclassical and LaguerreHahn $\LBP$ associated to generic closure
conditions of the chain of spectral transformations.
Keywords:Laurent orthogonal polynomials, Pastro polynomials, spectral transformations Category:33D45 

19. CMB 2000 (vol 43 pp. 496)
 Xu, Yuan

Harmonic Polynomials Associated With Reflection Groups
We extend Maxwell's representation of harmonic polynomials to $h$harmonics
associated to a reflection invariant weight function $h_k$. Let $\CD_i$,
$1\le i \le d$, be Dunkl's operators associated with a reflection group.
For any homogeneous polynomial $P$ of degree $n$, we prove the
polynomial $\xb^{2 \gamma +d2+2n}P(\CD)\{1/\xb^{2 \gamma +d2}\}$ is
a $h$harmonic polynomial of degree $n$, where $\gamma = \sum k_i$ and
$\CD=(\CD_1,\ldots,\CD_d)$. The construction yields a basis for
$h$harmonics. We also discuss selfadjoint operators acting on the
space of $h$harmonics.
Keywords:$h$harmonics, reflection group, Dunkl's operators Categories:33C50, 33C45 

20. CMB 1999 (vol 42 pp. 486)
 Sawyer, P.

Spherical Functions on $\SO_0(p,q)/\SO(p)\times \SO(q)$
An integral formula is derived for the spherical functions on the
symmetric space $G/K=\break
\SO_0(p,q)/\SO(p)\times \SO(q)$. This formula
allows us to state some results about the analytic continuation of
the spherical functions to a tubular neighbourhood of the
subalgebra $\a$ of the abelian part in the decomposition $G=KAK$.
The corresponding result is then obtained for the heat kernel of the
symmetric space $\SO_0(p,q)/\SO (p)\times\SO (q)$ using the Plancherel
formula.
In the Conclusion, we discuss how this analytic continuation can be
a helpful tool to study the growth of the heat kernel.
Categories:33C55, 17B20, 53C35 

21. CMB 1999 (vol 42 pp. 427)
 Berndt, Bruce C.; Chan, Heng Huat

Ramanujan and the Modular $j$Invariant
A new infinite product $t_n$ was introduced by S.~Ramanujan on the
last page of his third notebook. In this paper, we prove
Ramanujan's assertions about $t_n$ by establishing new connections
between the modular $j$invariant and Ramanujan's cubic theory of
elliptic functions to alternative bases. We also show that for
certain integers $n$, $t_n$ generates the Hilbert class field of
$\mathbb{Q} (\sqrt{n})$. This shows that $t_n$ is a new class
invariant according to H.~Weber's definition of class invariants.
Keywords:modular functions, the Borweins' cubic thetafunctions, Hilbert class fields Categories:33C05, 33E05, 11R20, 11R29 

22. CMB 1999 (vol 42 pp. 56)
 Elbert, Árpád; Siafarikas, Panayiotis D.

On the Square of the First Zero of the Bessel Function $J_\nu(z)$
Let $j_{\nu,1}$ be the smallest (first) positive zero of the Bessel
function $J_{\nu}(z)$, $\nu>1$, which becomes zero when $\nu$
approaches $1$. Then $j_{\nu,1}^{2}$ can be continued
analytically to $2<\nu<1$, where it takes on negative values. We
show that $j_{\nu,1}^{2}$ is a convex function of $\nu$ in the
interval $2<\nu\leq 0$, as an addition to an old result
[\'A.~Elbert and A.~Laforgia, SIAM J. Math. Anal. {\bf 15}(1984),
206212], stating this convexity for $\nu>0$. Also the
monotonicity properties of the functions $\frac{j_{\nu,1}^{2}}{4
(\nu+1)}$, $\frac{j_{\nu,1}^{2}}{4(\nu+1)\sqrt{\nu+2}}$ are
determined. Our approach is based on the series expansion of
Bessel function $J_{\nu}(z)$ and it turned out to be effective,
especially when $2<\nu<1$.
Category:33A40 

23. CMB 1998 (vol 41 pp. 86)