The explicit answer is known for your case if $3\alpha=-\delta$.

Glad to hear it. D.Karp is my former student and co-author. You may find many useful facts from his papers. May be start with log-convexity?

I am not sure that in the reference below there is an exact answer to the question.

Once I studied the variant of modified DFT with permutations of original rows or columns. In this case most properties are the same as for classical DFT (unitarian, explicit inversion). An interest was in spectral properties, they are different, this transform is known and useful in codes.

This is generalized Mittag-Leffler or Wright function.

Thank you. What I missed. Take such $x=a$ that $J_1(a)=0$. Then inf mod over all n will be zero, not so?

Is it a correct expression? It means we first take any x, then for this x find inf in n as function of x, and then take sup over x? Or it is better to change sup and inf?

An integral convergent test works for this problem, isn't it? It gives two-sided inequalities.

It seems can be generalized in the same way for higher degrees, $t^2 \to t^3, t^n$? What about general polynome $t^2 \to P(t)$ ?

Igor Rivin, your style is very nice. About the problem. As a result of Cauchy--Buniakowsky inequality will be integrals of squares of modules and they are all divergent, is not it? The same as integrals for $k=1$.

Dirk, is not it a question-- ask to find all such polynomes? And to generalize conditions for famous Airy and Fresnel functions--is not a motivation?

Was a missing prove really found somewhere?

I was always interesting to connect this function with some known say just for $s=1/2$. Its convolution square equals exp, if consider $\sum \frac{x^k}{(\sqrt{k!})}$. May somebody know something nontrivial for this function???