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Let $T (q) = \sum_{k = 1}^{\infty} d (k) q^{k}, | q | < 1,$ where $d (k)$ denotes the number of positive divisors of the natural number $k$ . We present monotonicity properties of functions defined in terms of $T$ . More specifically, we prove that $H (q) = T (q) - \frac{log (1 - q)}{log (q)}$ is strictly increasing on $(0, 1)$ , while $F (q) = \frac{1 - q}{q} H (q)$ is strictly decreasing on $(0, 1)$ . These results are then applied to obtain various inequalities, one of which states that the double inequality $α \frac{q}{1 - q} + \frac{log (1 - q)}{log (q)} < T (q) < β \frac{q}{1 - q} + \frac{log (1 - q)}{log (q)}, 0 < q < 1,$ holds with the best possible constant factors $α = γ$ and $β = 1$ . Here, $γ$ denotes Euler’s constant. This refines a result of Salem, who proved the inequalities...

Mean Value Theorems in q-calculus

Predrag Rajković, Miomir Stanković, Slađana Marinković (2002)

Matematički Vesnik

New inequalities for some special and $q$ -special functions.

Sellami, Mouna, Brahim, Kamel, Bettaibi, Néji (2007)

JIPAM. Journal of Inequalities in Pure & Applied Mathematics [electronic only]

On another extension of $q$ -Pfaff-Saalschütz formula

Mingjin Wang (2010)

Czechoslovak Mathematical Journal

In this paper we give an extension of $q$ -Pfaff-Saalschütz formula by means of Andrews-Askey integral. Applications of the extension are also given, which include an extension of $q$ -Chu-Vandermonde convolution formula and some other $q$ -identities.

On integral representations of $q$ -gamma and $q$ -beta functions

Alberto De Sole, Victor G. Kac (2005)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

We study $q$ -integral representations of the $q$ -gamma and the $q$ -beta functions. As an application of these integral representations, we obtain a simple conceptual proof of a family of identities for Jacobi triple product, including Jacobi's identity, and of Ramanujan's formula for the bilateral hypergeometric series.

On $q$ -summation and confluence

Lucia Di Vizio, Changgui Zhang (2009)

Annales de l’institut Fourier

This paper is divided in two parts. In the first part we study a convergent $q$ -analog of the divergent Euler series, with $q \in (0, 1)$ , and we show how the Borel sum of a generic Gevrey formal solution to a differential equation can be uniformly approximated on a convenient sector by a meromorphic solution of a corresponding $q$ -difference equation. In the second part, we work under the assumption $q \in (1, + \infty)$ . In this case, at least four different $q$ -Borel sums of a divergent power series solution of an irregular singular...

On some Feng Qi type $h$ -integral inequalities.

Krasniqi, Valmir, Shabani, Armend Sh. (2009)

International Journal of Open Problems in Computer Science and Mathematics. IJOPCM

On some Feng Qi type $q$ -integral inequalities.

Brahim, Kamel, Bettaibi, Néji, Sellami, Mouna (2008)

JIPAM. Journal of Inequalities in Pure & Applied Mathematics [electronic only]

On the $q$ -analogue of gamma functions and related inequalities.

Kim, Taekyun, Adiga, C. (2005)

JIPAM. Journal of Inequalities in Pure & Applied Mathematics [electronic only]

Polynomial Expansions for Solutions of Higher-Order Bessel Heat Equation in Quantum Calculus

Ben Hammouda, M.S., Nemri, Akram (2007)

Fractional Calculus and Applied Analysis

Mathematics Subject Class.: 33C10,33D60,26D15,33D05,33D15,33D90In this paper we give the q-analogue of the higher-order Bessel operators studied by I. Dimovski [3],[4], I. Dimovski and V. Kiryakova [5],[6], M. I. Klyuchantsev [17], V. Kiryakova [15], [16], A. Fitouhi, N. H. Mahmoud and S. A. Ould Ahmed Mahmoud [8], and recently by many other authors. Our objective is twofold. First, using the q-Jackson integral and the q-derivative, we aim at establishing some properties of this function with proofs...

Some inequalities for the $q$ -digamma function.

Mansour, Toufik, Shabani, Armend Sh. (2009)

JIPAM. Journal of Inequalities in Pure & Applied Mathematics [electronic only]

Some new Formulas involving $Γ_{q}$ Functions

Thomas Ernst (2007)

Rendiconti del Seminario Matematico della Università di Padova

The multiple gamma function and its q-analogue

Kimio Ueno, Michitomo Nishizawa (1997)

Banach Center Publications

We give an asymptotic expansion (the higher Stirling formula) and an infinite product representation (the Weierstrass product formula) of the Vignéras multiple gamma function by considering the classical limit of the multiple q-gamma function.

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