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Register a new userA note on the (h,q)-Zeta type function with weight alpha
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The objective of this paper is to derive symmetric property of (h,q)-Zeta function with weight alpha. By using this property, we give some interesting identities for (h,q)-Genocchi polynomials with weight alpha. As a result, our applications possess a number of interesting property which we state in this paper.
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We present several formulae for the large $t$ asymptotics of the Riemann zeta function $zeta(s)$, $s=sigma+i t$, $0leq sigma leq 1$, $t>0$, which are valid to all orders. A particular case of these results coincides with the classical results of Siegel. Using these formulae, we derive explicit representations for the sum $sum_a^b n^{-s}$ for certain ranges of $a$ and $b$. In addition, we present precise estimates relating this sum with the sum $sum_c^d n^{s-1}$ for certain ranges of $a, b, c, d$. We also study a two-parameter generalization of the Riemann zeta function which we denote by $Phi(u,v,beta)$, $uin mathbb{C}$, $vin mathbb{C}$, $beta in mathbb{R}$. Generalizing the methodology used in the study of $zeta(s)$, we derive asymptotic formulae for $Phi(u,v,beta)$.
In this paper we give the q-extension of Euler numbers which can be viewed as interpolating of the q-analogue of Euler zeta function ay negative integers, in the same way that Riemann zeta function interpolates Bernoulli numbers at negative integers. Finally we woll treat some identities of the q-extension of the euler numbers by using fermionic p-adic q-integration on Z_p.
In this paper we consider Dedekind type DC sums and prove receprocity laws related to DC sums.
We study the sum of the finite multiple harmonic $q$-series on $rtext{-}(r+1)$ indices at roots of unity with $r=1,2,3$. And we give the equivalent conditions of two conjectures regarding cyclic sums of finite multiple harmonic $q$-series on $1text{-}2text{-}3$ indices at roots of unity, posed recently by Kh. Pilehrood, T. Pilehrood and R. Tauraso.
Context. Statistical properties of HII region populations in disk galaxies yield important clues to the physics of massive star formation. Aims. We present a set of HII region catalogues and luminosity functions for a sample of 56 spiral galaxies in order to derive the most general form of their luminosity function. Methods. HII region luminosity functions are derived for individual galaxies which, after photometric calibration, are summed to form a total luminosity function comprising 17,797 HII regions from 53 galaxies. Results. The total luminosity function, above its lower limit of completeness, is clearly best fitted by a double power law with a significantly steeper slope for the high luminosity portion of the function. This change of slope has been reported in the literature for individual galaxies, and occurs at a luminosity of log L = 38.6pm0.1 (L in erg/s) which has been termed the Stromgren luminosity. A steep fall off in the luminosity function above log L = 40 is also noted, and is related to an upper limit to the luminosities of underlying massive stellar clusters. Detailed data are presented for the individual sample galaxies. Conclusions. The luminosity functions of HII regions in spiral galaxies show a two slope power law behaviour, with a significantly steeper slope for the high luminosity branch. This can be modelled by assuming that the high luminosity regions are density bounded, though the scenario is complicated by the inhomogeneity of the ionized interstellar medium. The break, irrespective of its origin, is of potential use as a distance indicator for disc galaxies.
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