Taekyun Kim, C. Adiga
In this, paper we obtain a q-analogue of a double inequality involving the Euler gamma function which was first proved geometrically by Alsina and Tomas and then analytically by Sandor
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We consider convexity and monotonicity properties for some functions related to the $q$-gamma function. As applications, we give a variety of inequalities for the $q$-gamma function, the $q$-digamma function $ψ_q(x)$, and the $q$-series. Among other consequences, we improve a result of Azler~and~Grinshpan about the zeros of the function $ψ_q(x)$. We use $q$-analogues for the Gauss multiplication formula to put in closed form members of some of our inequalities.
The fundamental aim of this paper is to describe q-Analogue of p-adic log gamma functions with weight alpha and beta. Moreover, we give relationship between p-adic q-log gamma funtions with weight (α, β) and q-extension of Genocchi numbers with weight alpha and beta and modified q Euler numbers with weight α
Analogues of Alder-Type Partition Inequalities for Fixed Perimeter Partitions
本文在 Fu 与 Tang 关于固定周长分拆的研究基础上，建立广义 Alder 型分拆不等式，并给出一个反向不等式。
In this paper we prove inequalities for multiplicative analogues of Diophantine exponents, similar to the ones known in the classical case. Particularly, we show that a matrix is badly approximable if and only if its transpose is badly approximable and establish some inequalities connecting multiplicative exponents with ordinary ones.
Let $F$ be a totally real field. For each ideal class $c$ of $F$ and each real embedding $ι$ of $F$, Hiroyuki Yoshida defined an invariant $X(c,ι)$ as a finite sum of log of Barnes' multiple gamma functions with some correction terms. Then the derivative value of the partial zeta function $ζ(s,c)$ has a canonical decomposition $ζ'(0,c)=\sum_ιX(c,ι)$, where $ι$ runs over all real embeddings of $F$. Yoshida studied the relation between $\exp(X(c,ι))$'s, Stark units, and Shimura's period symbol. Yoshida and the author also defined and studied the $p$-adic analogue $X_p(c,ι)$: In particular, we di…
Two analogues of the crank function are defined for overpartitions -- the first residual crank and the second residual crank. This suggests an exploration of crank functions defined for overpartitions whose parts are divisible by an arbitrary $d$. We examine the positive moments of these crank functions while varying $d$ and prove some inequalities.
We present a new proof of the "arithmetic" large sieve inequality, starting from the corresponding "harmonic" inequality, which is based on an amplification idea. We show that this also adapts to give some new sieve inequality for modular forms, where Hecke eigenvalues are thought as the analogues of the reductions of integers modulo primes.
Using the Jackson integral, we obtain the $q$-integral analogue of the Raabe type formulas for Barnes multiple Hurwitz-Lerch zeta functions and Barnes and Vardi's multiple gamma functions. Our results generalize $q$-integral analogue of the Raabe type formulas for the Hurwitz zeta functions and log gamma functions in [N. Kurokawa, K. Mimachi, and M. Wakayama, Jackson's integral of the Hurwitz zeta function, Rend. Circ. Mat. Palermo (2) 56 (2007), no. 1, 43--56]. During the proof we also obtain a new formula on the relationship between the higher and lower orders Hurwitz zeta functions.
In this paper we investigate the monotonicity properties related to the ratio of gamma functions, from which some related asymptotics and inequalities are established. Some special cases also confirm the conjectures of C.-P. Chen [Monotonicity properties, inequalities, and asymptotic expansions associated with the gamma function. Appl. Math. Comput. 283 (2016), 385--396.].
We derive two new analogues of a transformation formula of Ramanujan involving the Gamma and Riemann zeta functions present in the Lost Notebook. Both involve infinite series consisting of Hurwitz zeta functions and yield modular relations. As a special case of the first formula, we obtain an identity involving polygamma functions given by Andrew P. Guinand and as a limiting case of the second formula, we derive the transformation formula of Ramanujan.
Fixed Perimeter Analogues of Several Partition Results Related to Parity
该研究将奇偶性偏差不等式等分拆相关结果扩展至固定周长场景，推导了两类特殊分拆的递归公式，推广了经典分拆理论的相关结论。
In this paper we prove an analogue of the discrete spherical maximal theorem of Magyar, Stein, and Wainger, an analogue which concerns maximal functions associated to homogenous algebraic surfaces. Let $\mathfrak{p}$ be a homogenous polynomial in $n$ variables with integer coefficients of degree $d>1$. The maximal functions we consider are defined by \[ A_*f(y)=\sup_{N\geq1}\left|\frac{1}{r(N)}\sum_{\mathfrak{p}(x)=0;\,x\in[N]^n}f(y-x)\right|\] for functions $f:\mathbb{Z}^n\to\mathbb{C}$, where $[N]=\{-N,-N+1,...,N\}$ and $r(N)$ represents the number of integral points on the surface defined b…
In this paper we consider Kurepa's function $K(z)$ \cite{Kurepa_71}. We give some recurrent relations for Kurepa's function via appropriate sequences of rational functions and gamma function. Also, we give some inequalities for Kurepa's function $K(x)$ for positive values of $x$.
We construct the two-variable $p$-adic $q$-$L$-function which interpolates the generalized $q$-Bernoulli polynomials associated with primitive Dirichlet character $χ$. Indeed, this function is the $q$-extension of two-variable $p$-adic $L$-function due to Fox, corresponding to the case $q=1 .$ Finally, we give some $p$-adic integral representation for this two-variable $p$-adic $q$-$L$-function and derive to $q$-extension of the generalized formula of Diamond and Ferro and Greenberg for the two-variable $p$-adic $L$-function in terms of the $p$-adic gamma and $\log$ gamma function.
In this paper we consider alternating Kurepa's function $A(z)$ \cite{Petojevic_02}. We give some recurrent relations for alternating Kurepa's function via appropriate sequences of rational functions and gamma function. Also we give some inequalities for the real part of alternating Kurepa's function $A(x)$ for values of argument $x>-2$. The obtained results are analogous to results from \cite{Malesevc_04}.
From two q-summation formulas we deduce certain series expansion formulas involving the q-gamma function. With these formulas we can give q-analogues of series expansions for certain constants.
GCD and ABC inequalities for abelian varieties over function fields
该研究证明了特征0函数域上阿贝尔簇有理点的GCD不等式与几何abc猜想的特殊情形，采用三步证明模式，借助Xie-Yuan高度等工具解决了Buium的相关猜想。
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