Expand all Collapse all | Results 1 - 18 of 18 |
1. 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 |
2. CMB 2014 (vol 58 pp. 188)
Telescoping Estimates for Smooth Series We derive telescoping majorants and minorants for some classes
of series and give applications of these results.
Keywords:telescoping series, Stietjes constant, Hardy's formula, Stirling's formula Categories:26D15, 40A25, 97I30 |
3. CMB 2013 (vol 56 pp. 827)
Erratum to ``Quantum Limits of Eisenstein Series and Scattering States'' This paper provides an erratum to Y. N. Petridis,
N. Raulf, and M. S. Risager, ``Quantum Limits
of Eisenstein Series and Scattering States.'' Canad. Math. Bull., published
online 2012-02-03, http://dx.doi.org/10.4153/CMB-2011-200-2.
Keywords:quantum limits, Eisenstein series, scattering poles Categories:11F72, 8G25, 35P25 |
4. CMB 2013 (vol 56 pp. 673)
Diophantine Approximation for Certain Algebraic Formal Power Series in Positive Characteristic In this paper, we study rational approximations for certain algebraic power series over a finite field.
We obtain results for irrational elements of strictly positive degree
satisfying an equation of the type
\begin{equation}
\alpha=\displaystyle\frac{A\alpha^{q}+B}{C\alpha^{q}}
\end{equation}
where $(A, B, C)\in
(\mathbb{F}_{q}[X])^{2}\times\mathbb{F}_{q}^{\star}[X]$.
In particular,
we will give, under some conditions on the polynomials $A$, $B$
and $C$, well approximated elements satisfying this equation.
Keywords:diophantine approximation, formal power series, continued fraction Categories:11J61, 11J70 |
5. CMB 2012 (vol 56 pp. 814)
Quantum Limits of Eisenstein Series and Scattering States We identify the quantum limits of scattering states
for the modular surface. This is obtained through the study of quantum
measures of non-holomorphic Eisenstein series away from the critical
line. We provide a range of stability for the quantum unique
ergodicity theorem of Luo and Sarnak.
Keywords:quantum limits, Eisenstein series, scattering poles Categories:11F72, 58G25, 35P25 |
6. CMB 2011 (vol 56 pp. 258)
The Smallest Pisot Element in the Field of Formal Power Series Over a Finite Field Dufresnoy and Pisot characterized the smallest
Pisot number of degree $n \geq 3$ by giving explicitly its minimal
polynomial. In this paper, we translate Dufresnoy and Pisot's
result to the Laurent series case.
The
aim of this paper is to prove that the minimal polynomial
of the smallest Pisot element (SPE) of degree $n$ in the field of
formal power series over a finite field
is given by $P(Y)=Y^{n}-\alpha XY^{n-1}-\alpha^n,$ where $\alpha$
is the least element of the finite field $\mathbb{F}_{q}\backslash\{0\}$
(as a finite total ordered set). We prove that the sequence of
SPEs of degree $n$ is decreasing and converges to $\alpha X.$
Finally, we show how to obtain explicit continued fraction
expansion of the smallest Pisot element over a finite field.
Keywords:Pisot element, continued fraction, Laurent series, finite fields Categories:11A55, 11D45, 11D72, 11J61, 11J66 |
7. 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 |
8. CMB 2011 (vol 55 pp. 26)
A Mahler Measure of a $K3$ Surface Expressed as a Dirichlet $L$-Series We present another example of a $3$-variable polynomial defining a $K3$-hypersurface
and having a logarithmic Mahler measure expressed in terms of a Dirichlet
$L$-series.
Keywords:modular Mahler measure, Eisenstein-Kronecker series, $L$-series of $K3$-surfaces, $l$-adic representations, LivnÃ© criterion, Rankin-Cohen brackets Categories:11, 14D, 14J |
9. CMB 2011 (vol 55 pp. 60)
Extension of Some Theorems of W. Schwarz In this paper, we prove that a non--zero power series $F(z)\in\mathbb{C}
[\mkern-3mu[ z]\mkern-3mu]
$
satisfying $$F(z^d)=F(z)+\frac{A(z)}{B(z)},$$ where $d\geq 2$, $A(z),B(z)\in\mathbb{C}[z]$
with $A(z)\neq 0$ and $\deg A(z),\deg B(z) Keywords:functional equations, transcendence, power series Categories:11B37, 11J81 |
10. CMB 2009 (vol 52 pp. 481)
Some Infinite Products of Ramanujan Type In his ``lost'' notebook, Ramanujan stated two results, which are equivalent to the identities
\[
\prod_{n=1}^{\infty} \frac{(1-q^n)^5}{(1-q^{5n})}
=1-5\sum_{n=1}^{\infty}\Big( \sum_{d \mid n} \qu{5}{d} d \Big) q^n
\]
and
\[
q\prod_{n=1}^{\infty} \frac{(1-q^{5n})^5}{(1-q^{n})}
=\sum_{n=1}^{\infty}\Big( \sum_{d \mid n} \qu{5}{n/d} d \Big) q^n.
\]
We give several more identities of this type.
Keywords:Power series expansions of certain infinite products Categories:11E25, 11F11, 11F27, 30B10 |
11. CMB 2009 (vol 52 pp. 627)
On $L^{1}$-Convergence of Fourier Series under the MVBV Condition Let $f\in L_{2\pi }$ be a real-valued even function with its Fourier series $%
\frac{a_{0}}{2}+\sum_{n=1}^{\infty }a_{n}\cos nx,$ and let
$S_{n}(f,x) ,\;n\geq 1,$ be the $n$-th partial sum of the Fourier series. It
is well known that if the nonnegative sequence $\{a_{n}\}$ is decreasing and
$\lim_{n\rightarrow \infty }a_{n}=0$, then%
\begin{equation*}
\lim_{n\rightarrow \infty }\Vert f-S_{n}(f)\Vert _{L}=0
\text{ if
and only if }\lim_{n\rightarrow \infty }a_{n}\log n=0.
\end{equation*}%
We weaken the monotone condition in this classical result to the so-called
mean value bounded variation (MVBV) condition. The generalization of the
above classical result in real-valued function space is presented as a
special case of the main result in this paper, which gives the $L^{1}$%
-convergence of a function $f\in L_{2\pi }$ in complex space. We also give
results on $L^{1}$-approximation of a function $f\in L_{2\pi }$ under the
MVBV condition.
Keywords:complex trigonometric series, $L^{1}$ convergence, monotonicity, mean value bounded variation Categories:42A25, 41A50 |
12. CMB 2008 (vol 51 pp. 3)
13. 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 |
14. CMB 2007 (vol 50 pp. 11)
van der Pol Expansions of L-Series We provide concise series representations for various
L-series integrals. Different techniques are needed below and above
the abscissa of absolute convergence of the underlying L-series.
Keywords:Dirichlet series integrals, Hurwitz zeta functions, Plancherel theorems, L-series Categories:11M35, 11M41, 30B50 |
15. CMB 2006 (vol 49 pp. 256)
A Bernstein--Walsh Type Inequality and Applications A Bernstein--Walsh type inequality for $C^{\infty }$ functions of several
variables is derived, which then is applied to obtain analogs and
generalizations of the following classical theorems: (1) Bochnak--Siciak
theorem: a $C^{\infty }$\ function on $\mathbb{R}^{n}$ that is real
analytic on every line is real analytic; (2) Zorn--Lelong theorem: if a
double power series $F(x,y)$\ converges on a set of lines of positive
capacity then $F(x,y)$\ is convergent; (3) Abhyankar--Moh--Sathaye theorem:
the transfinite diameter of the convergence set of a divergent series is
zero.
Keywords:Bernstein-Walsh inequality, convergence sets, analytic functions, ultradifferentiable functions, formal power series Categories:32A05, 26E05 |
16. 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 |
17. CMB 1999 (vol 42 pp. 198)
Commutators and Analytic Dependence of Fourier-Bessel Series on $(0,\infty)$ In this paper we study the boundedness of the commutators $[b,
S_n]$ where $b$ is a $\BMO$ function and $S_n$ denotes the $n$-th
partial sum of the Fourier-Bessel series on $(0,\infty)$.
Perturbing the measure by $\exp(2b)$ we obtain that certain
operators related to $S_n$ depend analytically on the functional
parameter $b$.
Keywords:Fourier-Bessel series, commutators, BMO, $A_p$ weights Category:42C10 |
18. 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 |