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1. CMB Online first

Wirths, Karl Joachim
 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 formulaCategories:26D15, 40A25, 97I30

2. CMB 2013 (vol 56 pp. 827)

Petridis, Yiannis N.; Raulf, Nicole; Risager, Morten S.
 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 polesCategories:11F72, 8G25, 35P25

3. CMB 2013 (vol 56 pp. 673)

Ayadi, K.; Hbaib, M.; Mahjoub, F.
 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 $$\alpha=\displaystyle\frac{A\alpha^{q}+B}{C\alpha^{q}}$$ 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 fractionCategories:11J61, 11J70

4. CMB 2012 (vol 56 pp. 814)

Petridis, Yiannis N.; Raulf, Nicole; Risager, Morten S.
 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 polesCategories:11F72, 58G25, 35P25

5. CMB 2011 (vol 56 pp. 258)

Chandoul, A.; Jellali, M.; Mkaouar, M.
 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 fieldsCategories:11A55, 11D45, 11D72, 11J61, 11J66

6. CMB 2011 (vol 55 pp. 571)

Miller, A. R.; Paris, R. B.
 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 transformationCategories:33C15, 33C20

7. CMB 2011 (vol 55 pp. 26)

Bertin, Marie José
 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 bracketsCategories:11, 14D, 14J

8. CMB 2011 (vol 55 pp. 60)

Coons, Michael
 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 $\deg A(z),\deg B(z) Keywords:functional equations, transcendence, power seriesCategories:11B37, 11J81 9. CMB 2009 (vol 52 pp. 627) Yu, Dan Sheng; Zhou, Ping; Zhou, Song Ping  On$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 variationCategories:42A25, 41A50 10. CMB 2009 (vol 52 pp. 481) Alaca, Ay\c{s}e; Alaca, \c{S}aban; Williams, Kenneth S.  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 productsCategories:11E25, 11F11, 11F27, 30B10 11. CMB 2008 (vol 51 pp. 3) 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 integralCategories:11B65, 33D15 13. CMB 2007 (vol 50 pp. 11) Borwein, David; Borwein, Jonathan  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-seriesCategories:11M35, 11M41, 30B50 14. CMB 2006 (vol 49 pp. 256) Neelon, Tejinder  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 seriesCategories:32A05, 26E05 15. CMB 2005 (vol 48 pp. 147) Väänänen, Keijo; Zudilin, Wadim  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,$q$-seriesCategories:11J82, 33D15 16. CMB 1999 (vol 42 pp. 198) Guadalupe, José J.; Pérez, Mario; Varona, Juan L.  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,$A_p$weightsCategory:42C10 17. CMB 1998 (vol 41 pp. 86) Lubinsky, D. S.  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\$-exponentialsCategories:33D05, 11A55, 11K70