# How to prove that $sum_{m=0}^{infty} { Gamma{(1+2m)/alpha}over Gamma(1/2+m)} { (-t^2/4)^{m}over m !} ge (alpha/2)^{3}exp(-t^{2}/4)$

MathOverflow Asked on January 3, 2022

I would love to prove the following inequality
$${1over sqrt{pi} } sum_{m=0}^{infty} Gamma{(1+2m)/alpha} { (-t^2)^{m}over (2m) !}=$$
$$sum_{m=0}^{infty} { Gamma{(1+2m)/alpha}over Gamma(1/2+m)} { (-t^2/4)^{m}over m !} ge (alpha/2)^{3}exp(-t^{2}/4)$$
$$1, $$t>0$$,
The question is connected to the other question I asked and got no answer for Prove $int_{0}^{infty} cos(omega x) exp(-x^{alpha}) , {rm d} x ge {alpha^2 sqrt{pi} over 8} exp left( -frac{omega^2}{4} right)$

## Related Questions

### Closure of the product of subfunctors

1  Asked on December 8, 2021 by nate-gallup

### Small-$r$ asymptotics of an integral of $1/log ^alpha r$

1  Asked on December 8, 2021 by yongpan-huang

### Does $mathcal{A}otimesmathbb{C}(t)congmathcal{D}otimesmathbb{C}(t)$ imply an isomorphism of Lie algebras?

1  Asked on December 8, 2021 by solver6

### When does $det begin{pmatrix} A & X \ X^T & A end{pmatrix} = (det A)^2 + (det X)^2$?

1  Asked on December 8, 2021

### When does cohomology of a pro-algebraic group commute with filtered colimits of coefficients?

1  Asked on December 8, 2021 by patrick-elliott

### Bounded weak and weak-$star$ topologies and metrics

0  Asked on December 8, 2021 by jorge-e-cardona

### Is semi-simplicity of Galois representations local?

0  Asked on December 8, 2021

### Maximum eigenvalue of a covariance matrix of Brownian motion

2  Asked on December 6, 2021 by weiqiang-yang

### Identity theorem in $p$-adic geometry/analysis

1  Asked on December 6, 2021

### Updates to Stanley’s 1999 survey of positivity problems in algebraic combinatorics?

1  Asked on December 5, 2021 by sam-hopkins

### Construction of $K(Gal(bar{k}/k), 1)$

0  Asked on December 5, 2021 by jog

### Forcing as a tool to prove theorems

9  Asked on December 3, 2021

### The determinant of the sum of normal matrices

1  Asked on December 3, 2021 by denis-serre

### Block matrices and their determinants

1  Asked on December 3, 2021 by t-amdeberhan

### Extending submodular functions from a sublattice

1  Asked on December 3, 2021 by darij-grinberg

### Regularity properties of Minakshisundaram–Pleijel zeta function

0  Asked on December 3, 2021 by alephbeth

### On proving the absence of limit cycles in a dynamical system

1  Asked on December 3, 2021 by norregaard

### Computing the infinite dimensional Lebesgue measure of “cubes”

0  Asked on December 3, 2021

### Why do Todd classes appear in Grothendieck-Riemann-Roch formula?

4  Asked on December 1, 2021 by ilya-nikokoshev