Expand all Collapse all | Results 1 - 21 of 21 |
1. CMB Online first
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 |
2. CMB Online first
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 Whipple-Sears 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 |
3. CMB 2011 (vol 55 pp. 571)
A Generalised Kummer-Type Transformation for the ${}_pF_p(x)$ Hypergeometric Function In a recent paper, Miller derived a Kummer-type
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 well-known Kummer transformation for the
confluent hypergeometric function corresponding to $p=1$.
Keywords:generalised hypergeometric series, Kummer transformation Categories:33C15, 33C20 |
4. CMB 2010 (vol 54 pp. 538)
On the Horizontal Monotonicity of $|\Gamma(s)|$
Writing $s = \sigma + it$ for a complex variable, it is proved
that the modulus of the gamma
function, $|\Gamma(s)|$, is strictly monotone increasing with
respect to $\sigma$ whenever
$|t| > 5/4$. It is also shown that this result is false for $|t|
\leq 1$.
Keywords:Gamma function, modulus, monotonicity Category:33B15 |
5. CMB 2009 (vol 52 pp. 583)
Computing Polynomials of the Ramanujan $t_n$ Class Invariants We compute the minimal polynomials of the Ramanujan values $t_n$,
where $n\equiv 11 \mod 24$, using the Shimura reciprocity law.
These polynomials can be used for defining the Hilbert class field
of the imaginary quadratic field $\mathbb{Q}(\sqrt{-n})$ and have
much smaller coefficients than the Hilbert polynomials.
Categories:11R29, 33E05, 11R20 |
6. CMB 2008 (vol 51 pp. 561)
Expansion of the Riemann $\Xi$ Function in Meixner--Pollaczek Polynomials In this article we study in detail the expansion of the Riemann
$\Xi$ function in Meixner--Pollaczek polynomials. We obtain explicit
formulas, recurrence relation and asymptotic expansion for the
coefficients and investigate the zeros of the partial sums.
Categories:41A10, 11M26, 33C45 |
7. CMB 2008 (vol 51 pp. 627)
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 |
8. CMB 2007 (vol 50 pp. 547)
Inverse Laplace Transforms Encountered in Hyperbolic Problems of Non-Stationary Fluid-Structure Interaction |
Inverse Laplace Transforms Encountered in Hyperbolic Problems of Non-Stationary Fluid-Structure 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
non-stationary fluid-structure interaction, and as such are of
interest to researchers working in the area.
Categories:44A10, 44A20, 33C10, 40A30, 74F10, 76Q05 |
9. CMB 2007 (vol 50 pp. 284)
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 |
10. CMB 2005 (vol 48 pp. 382)
Uniform Estimates of Ultraspherical Polynomials of Large Order In this paper we prove the sharp inequality
$$ |P_n^{(s)}(x)|\leq
P_n^{(s)}(1)\bigl(|x|^n +\frac{n-1}{2 s+1}(1-|x|^n)\bigr),$$
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 |
11. CMB 2005 (vol 48 pp. 147)
Baker-Type 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 non-zero 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\'e-type 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 |
12. CMB 2002 (vol 45 pp. 567)
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 ways---we 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 |
13. CMB 2002 (vol 45 pp. 436)
The Spherical Functions Related to the Root System $B_2$ In this paper, we give an integral formula for the eigenfunctions of
the ring of differential operators related to the root system $B_2$.
Categories:43A90, 22E30, 33C80 |
14. CMB 2002 (vol 45 pp. 257)
Modular Forms Associated to Theta Functions We use the theory of Jacobi-like 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 |
15. CMB 2001 (vol 44 pp. 337)
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 1-1-periodic $q$-closure condition
leads to the $\LBP$ introduced by Pastro. We introduce classes of
semi-classical and Laguerre-Hahn $\LBP$ associated to generic closure
conditions of the chain of spectral transformations.
Keywords:Laurent orthogonal polynomials, Pastro polynomials, spectral transformations Category:33D45 |
16. CMB 2000 (vol 43 pp. 496)
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 $|\xb|^{2 \gamma +d-2+2n}P(\CD)\{1/|\xb|^{2 \gamma +d-2}\}$ 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 self-adjoint operators acting on the
space of $h$-harmonics.
Keywords:$h$-harmonics, reflection group, Dunkl's operators Categories:33C50, 33C45 |
17. CMB 1999 (vol 42 pp. 427)
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 theta-functions, Hilbert class fields Categories:33C05, 33E05, 11R20, 11R29 |
18. CMB 1999 (vol 42 pp. 486)
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 |
19. CMB 1999 (vol 42 pp. 56)
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),
206--212], 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 |
20. CMB 1998 (vol 41 pp. 86)
On \lowercase{$q$}-exponential functions for \lowercase{$|q| =1$} We discuss the $q$-exponential functions $e_q$, $E_q$ for $q$ on the unit
circle, especially their continuity in $q$, and analogues of the limit
relation $ \lim_{q\rightarrow 1}e_q((1-q)z)=e^z$.
Keywords:$q$-series, $q$-exponentials Categories:33D05, 11A55, 11K70 |
21. CMB 1997 (vol 40 pp. 276)
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 |