Related papers: The multiple gamma functions and the multiple q-ga…

We consider differences between $\log \Gamma(x)$ and truncations of certain classical asymptotic expansions in inverse powers of $x-\lambda$ whose coefficients are expressed in terms of Bernoulli polynomials $B_n(\lambda)$, and we obtain…

We present two integral representations of the logarithm of the Glaisher-Kinkelin constant. The calculations are based on definite integral expressions of $\log\Gamma(x)$, $\Gamma$ being the usual Gamma function, due respectively to F\'eaux…

We study the equation $M_\Psi(z+1)=\frac{-z}{\Psi(-z)}M_\Psi(z), M_\Psi(1)=1$ defined on a subset of the imaginary line and where $\Psi$ is a negative definite functions. Using the Wiener-Hopf method we solve this equation in a two terms…

Stirling's formula, the asymptotic expansion of $n!$ for $n$ large, or of $\Gamma(z)$ for $z\to \infty$, is derived directly from the recursion equation $\Gamma(z+1) =z \Gamma(s)$ and the normalization condition $\Gamma ({1/2})…

A representation for the Riemann zeta function valid for arbitrary complex $s=\sigma+it$ is $\zeta(s)=\sum_{n=0}^\infty A(n,s)$, where \[A(n,s)=\frac{2^{-n-1}}{1-2^{1-s}} \sum_{k=0}^n \left(\!\begin{array}{c}n\\k\end{array}\!\right)…

We show that the Wigner equations describing the continuous spin representations can be obtained as a limit of massive higher-spin field equations. The limit involves a suitable scaling of the wave function, the mass going to zero and the…

We consider the asymptotic behavior of the incomplete gamma functions gamma(-a,-z) and Gamma(-a,-z) as a goes to infinity. Uniform expansions are needed to describe the transition area z~a in which case error functions are used as main…

Under the Riemann Hypothesis, we prove for any natural number $r$ there exist infinitely many large natural numbers $n$ such that $(\gamma_{n+r}-\gamma_n)/(2\pi /\log \gamma_n) > r + \Theta\sqrt{r}$ and $(\gamma_{n+r}-\gamma_n)/(2\pi /\log…

We present new addition formulae for the Weierstrass functions associated with a general elliptic curve. We prove the structure of the formulae in n-variables and give the explicit addition formulae for the 2- and 3-variable cases. These…

This paper considers some integrals where the integrand comprises the log gamma function or the digamma function multiplied by exponential or trigonometric functions.

This paper presents expressions for gamma values at rational points with the denominator dividing 24 or 60. These gamma values are expressed in terms of 10 distinct gamma values and rational powers of $\pi$ and a few real algebraic numbers.…

Letting $(t_n)$ denote the Thue-Morse sequence with values $0, 1$, we note that the Woods-Robbins product $$ \prod_{n \geq 0} \left(\frac{2n+1}{2n+2}\right)^{(-1)^{t_n}} = 2^{-1/2} $$ involves a rational function in $n$ and the $\pm 1$…

We give a precise formula for the value of the canonical Green's function at a pair of Weierstrass points on a hyperelliptic Riemann surface. Further we express the 'energy' of the Weierstrass points in terms of a spectral invariant…

We obtain several expansions for $\zeta(s)$ involving a sequence of polynomials in $s$, denoted in this paper by $\alpha_k(s)$. These polynomials can be regarded as a generalization of Stirling numbers of the first kind and our identities…

This paper analyzes over 30 types of q-series and the asymptotic behavior of their expansions. A method is described for deriving further asymptotic formulas using convolutions of generating functions with subexponential growth. All…

In the paper, the authors establish an inequality involving exponential functions and sums, introduce a ratio of many gamma functions, discuss properties, including monotonicity, logarithmic convexity, (logarithmically) complete…

Answering a question of Geoff Robinson, we compute the large n limiting proportion of i(n,q)/q^[n^2/2], where i(n,q) denotes the number of involutions in GL(n,q). We give similar results for the finite unitary, symplectic, and orthogonal…

This paper derives a way to express differentiable complex-valued functions as the sum of powers of $(1-e^{\lambda x})$, where $\lambda\in\mathbb{R}$, with an explicit formula for the remainder. This formulation is then used to associate an…

This paper shows that for a given irreducible representation $\rho$ of $\Gamma/\Gamma_1$, the two functions dim($M_k(\Gamma_1,\rho)$) and dim($S_k(\Gamma_1,\rho)$) of $k$ are almost linear functions.

We present details of logically simplest integral sufficient for deducing the Stirling asymptotic formula for n!. It is the Newton integral, defined as the difference of values of any primitive at the endpoints of the integration interval.…