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First, some notation. I'll write $f(x)=o(g(x))$ if $\lim_{x\to\infty} \left|\frac{f(x)}{g(x)}\right|=0$. I'll also write $g(x)=\omega(f(x))$ if $f(x)=o(g(x))$, i.e. $\limsup_{x\to\infty} \left|\frac{g(x)}{f(x)}\right|=\infty$. I'll say $f(x)\sim g(x)$ as $x\to\infty$ if $f(x)=g(x)+o(g(x))$ for all large enough $x$, i.e. $f(x)/g(x)=1+o(1)$ for all large enough $x$.

I recently posted a question on MathOverflow asking whether $$\sum_{k\ge 0}\frac{x^k}{\Gamma(1+a(k))}\sim\int_0^\infty \frac{x^t}{\Gamma(1+a(t))}\,dt\tag{1}\label{1}$$ as $x\to\infty$, given that $a(t)\in\omega\left(\frac{t}{\log t}\right)\cap C^\infty[0,\infty)$ as well as being non-negative and monotonically increasing. The answer given provided the a counter-example. In this question, I am asking about what necessary and sufficient conditions must be placed on the function $a(t)$ so that $(\ref{1})$ holds.

I know this relationships hold true when $a(k):=k$ (see [2], [3]), in which case the series is simply $e^x$, as well as when $a(k):=2k$, in which case the series is $\cosh(\sqrt{x})$.

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  • $\begingroup$ "what necessary and sufficient conditions" ... Almost all questions of the form "What is A?" are meaningless -- because "A is A" is always an answer here. Asking such a "what is" question, one should say in what terms the object A is to be described (unless this is naturally and commonly understood). In this case, the terms of a desired description should certainly be provided (even though I suspect this can hardly be done, and therefore a reasonable answer can hardly be provided here.) $\endgroup$
    – Iosif Pinelis
    Commented Jan 5, 2023 at 15:34

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