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1. CMB Online first

Benamar, H.; Chandoul, A.; Mkaouar, M.
 On the continued fraction expansion of fixed period in finite fields The Chowla conjecture states that, if $t$ is any given positive integer, there are infinitely many prime positive integers $N$ such that $\operatorname{Per} (\sqrt{N})=t$, where $\operatorname{Per} (\sqrt{N})$ is the period length of the continued fraction expansion for $\sqrt{N}$. C. Friesen proved that, for any $k\in \mathbb{N}$, there are infinitely many square-free integers $N$, where the continued fraction expansion of $\sqrt{N}$ has a fixed period. In this paper, we describe all polynomials $Q\in \mathbb{F}_q[X]$ for which the continued fraction expansion of $\sqrt {Q}$ has a fixed period, also we give a lower bound of the number of monic, non-squares polynomials $Q$ such that $\deg Q= 2d$ and $Per \sqrt {Q}=t$. Keywords:continued fractions, polynomials, formal power seriesCategories:11A55, 13J05

2. CMB Online first

Hu, Pei-Chu; Li, Bao Qin
 A simple proof and strengthening of a uniqueness theorem for L-functions We give a simple proof and strengthening of a uniqueness theorem for functions in the extended Selberg class. Keywords:meromorphic function, Dirichlet series, L-function, zero, order, uniquenessCategories:30B50, 11M41

3. CMB 2015 (vol 58 pp. 548)

 Higher Moments of Fourier Coefficients of Cusp Forms Let $S_{k}(\Gamma)$ be the space of holomorphic cusp forms of even integral weight $k$ for the full modular group $SL(2, \mathbb{Z})$. Let $\lambda_f(n)$, $\lambda_g(n)$, $\lambda_h(n)$ be the $n$th normalized Fourier coefficients of three distinct holomorphic primitive cusp forms $f(z) \in S_{k_1}(\Gamma), g(z) \in S_{k_2}(\Gamma), h(z) \in S_{k_3}(\Gamma)$ respectively. In this paper we study the cancellations of sums related to arithmetic functions, such as $\lambda_f(n)^4\lambda_g(n)^2$, $\lambda_g(n)^6$, $\lambda_g(n)^2\lambda_h(n)^4$, and $\lambda_g(n^3)^2$ twisted by the arithmetic function $\lambda_f(n)$. Keywords:Fourier coefficients of automorphic forms, Dirichlet series, triple product $L$-function, Perron's formulaCategories:11F30, 11F66

4. CMB Online first

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 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 seriesCategory:33D20

5. CMB 2014 (vol 58 pp. 188)

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

6. 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

7. 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

8. 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

9. 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

10. 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

11. 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

12. 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 $\deg A(z),\deg B(z) Keywords:functional equations, transcendence, power seriesCategories:11B37, 11J81 13. CMB 2009 (vol 52 pp. 627) Yu, Dan Sheng; Zhou, Ping; Zhou, Song Ping  On$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 14. 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 15. CMB 2008 (vol 51 pp. 3)  Alaca, Ay\c{s}e; Alaca, \c{S}aban; Williams, Kenneth S. 16. 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 17. 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 18. 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 19. 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 20. 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 21. 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
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