## Is there a closed form expression/series expansion for ∫ϵ+i∞ϵ−i∞eaz+b2z2Γ(z)Γ(1−z)dz\int_{\epsilon-i\infty}^{\epsilon+i\infty} e^{az+b^2z^2}\Gamma(z)\Gamma(1-z)dz ?

I’ve been trying to find a closed form expression/series expansion for the following integral without success: F(a,b)=∫ϵ+i∞ϵ−i∞eaz+b2z2Γ(z)Γ(1−z)dz where a,b,ϵ>0. Any input is greatly appreciated! Answer Hi dp, Using the approach I mentioned above, that is splitting the csc(πz) term into its pole part 1πz and the rest, I get an exact answer for the pole … Read more

## Xi Function on Critical Strip – Mellin Transform

Story I’m trying to prove following identity ∫∞0Ξ(t)t2+14cos(xt)dt=12π(e12x−2e−12xψ(e−2x)) where ψ(x)=∞∑n=1e−n2πx and Ξ(t)=ξ(12+it) is the xi function on the critical line. Problem I only don’t understand the following equality −14i√y∫12+i∞12−i∞Γ(12s)π−12sζ(s)ysds=−π√yψ(1y2)+12π√y. Where does latter summand 12π√y come from? When I write (1y)−s and substitute with s=2w I only get the first summand. We know that by the … Read more

## Proving two inequalities involving the gamma and digamma functions

I’m having trouble proving the following inequality: ∀p>1∀m≥0m2Γ(2mp)Γ(2mq)Γ(2m+2p)Γ(2m+2q)≥14p2(p−1)2p−2, where as usual q=pp−1. In fact, it seems clear from Mathematica that for a fixed p, the LHS is a decreasing function of m (strictly unless p=2, in which case it’s constant). The RHS can be seen to be the limit as m→∞. I actually only care … Read more

## Do the roots of this equation involving two Euler products all reside on the critical line?

This question loosely builds on the second part of this one. Take the Riemann ξ-function: ξ(s)=12s(s−1)π−s2Γ(s2)ζ(s). Numerical evidence suggests that for all a∈R, the zeros of: g(s,a):=ξ(a+s)±ξ(a+1−s) all reside on the critical line ℜ(s)=12. The graph below illustrates the claim for −2<a<2 (and ±=+). Each line shows ℑ(s) of a zero for g(s,a) at ℜ(s)=12. … Read more

## An integral identity relate to the Gamma function or the Beta function

I encountered the following identity in a paper on number theory, $$\int_{-\infty}^{\infty}\frac{dW}{(W+i)^{\frac{3}{2}}(W^2+1)^s}=\frac{e^\frac{-3\pi i}{4}\sqrt{2}\pi \Gamma(2s+\frac{1}{2})}{2^{2s}\Gamma(s+\frac{3}{2})\Gamma(s)},$$ with $Re(s) > 0$ and $i=\sqrt{-1}$. Since the author did not give the proof for this, maybe it is “well-known”, but I failed to gave a proof. Note that the right hand side looks like the Beta function with some multiplier, … Read more

## Hankel determinant of incomplete gamma functions

I have some expressions that involve Hankel determinants of incomplete gamma functions. They are of the (r×r form) I’d like to evaluate these determinants. Elementary operations help, but these determinants are so ill-conditioned (large n, r) that I have yet to find any technique (equilibration, etc) that provides sufficient stability. There is a body of … Read more

## q-Pochhammer Symbol Identity

Is this identity or an equivalent one already referenced in the litterature? In particular, is it even true? (−1;e−4π)2∞(e−2π;e−2π)4∞=32π3(√2−1)√214+234Γ4(14)≃4,030103529… It appears in relation to a particular elliptic function. Similar identities also arise in this post Thanks in advance, Answer The equality is indeed correct. It follows from identities in Ramanujan’s notebook. First notice that (−1;e−4π)2∞=2(−e−4π;e−4π)2∞, … Read more

## The Riemann Zeta Function summing over the Gamma Function

Has anyone studied a function of the form $$\eta(s) = \sum_{n=1}^{\infty} \frac{1}{\Gamma(n)^{s}} = \sum_{n=0}^{\infty}\frac{1}{k!^s}$$ This series is appearing in my research on the volumetric properties of infinite families of polytopes and I am not sure how to evaluate the sum, or if this is a well-known function other than the fact that $\eta(2) = I_{0}(2)$, … Read more

## A relation between the Gamma function and the Mobius function?

It is well known how altering the integral for the Gamma function: $$\displaystyle \Gamma(s) = \int_0^\infty t^{s-1} e^{-t}\,dt$$ through substituting $t=nx$, $$\displaystyle \Gamma(s)\frac{1}{n^s} = \int_0^\infty x^{s-1} e^{-n\,x}\,dx$$ and summing both sides, will “give birth” to the $\zeta$ function for $\Re(s) \gt 1$. $$\displaystyle \Gamma(s)\zeta(s)=\Gamma(s)\sum_{n=1}^{\infty}\frac{1}{n^s} = \int_0^\infty x^{s-1} \sum_{n=1}^{\infty}e^{-n\,x}\,dx$$ This can be extended further by introducing … Read more

## how understand periodicity in a combination of power, gamma and zeta functions?

Riemann’s functional equation may be written: ζ(s)ζ(1−s)=2sπs−1sin(πs2)Γ(1−s) and so by symmetry: ζ(1−s)ζ(s)=21−sπ−scos(πs2)Γ(s) multiplying the two versions gives Euler’s reflection formula. now define a function Ψ(s)=(2π)−sΓ(s)ζ2(s) so that the result of dividing (1) by (2) is expressed as: Ψ(s)Ψ(1−s)=tanπs2 question is there any intuitively appealing “explanation” for this periodicity? Answer AttributionSource : Link , Question Author … Read more