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

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

integrating with respect to parameters in beta function

I would like to evaluate an integral: $$\int_t^1\frac{1}{B(1+s\phi,1+\phi(1-s))}p^{s\phi}(1-p)^{\phi(1-s)}ds,$$ where $B(a,b)$ is a beta function and $p\in(0,1)$ and $\phi>0$ are some parameters. Notice that the integration is with respect to $s$, which appears both in the exponent and in the beta function. This is integrating over a family of pdfs of beta distributions, indexed by the … Read more