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Integrating $\int_0^\infty \sqrt x e^{-4/3x^{3/2}}\left(\int_0^x \operatorname{Ai}(t)dt\int_0^x \operatorname{Bi}(t)dt\right)dx$

$\DeclareMathOperator{\Ai}{\mathrm{Ai}} \DeclareMathOperator{\Bi}{\mathrm{Bi}}$This is not a full answer (yet). As commented by @Timothy, the problem can be rewritten using modified Bessel functions ...
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What are the properties of umbra with moments $\{1,1/2,1/3,1/4,1/5,...\}$?

I'll sketch here some of the general Sheffer Appell-umbral calculus relationships between the Bernoulli numbers / polynomials and the reciprocal (natural) numbers / polynomials. Perhaps when I have ...
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Where or what is the general formula for the $n$th derivative of the power-exponential function $x^x$?

The Stirling numbers of the first kind $s(n,k)$ for $n\ge k\ge0$ can be analytically generated by \begin{equation*}%\label{1st-stirl-gen-funct} \frac{[\ln(1+t)]^k}{k!}=\sum_{n=k}^\infty s(n,k)\frac{t^...
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Identity involving Stirling number of the second kind

The first formula in Section 24.1.4.I.B in Handbook of mathematical functions with formulas, graphs, and mathematical tables, Dover edition, 1965 (Library of Congress Catalog Card Number: 65.12253) by ...
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Solutions of complex linear difference equations

Look for a solution of the form $f(z)=e^{\lambda z}$. Plugging this to your equation, you obtain that $\lambda$ must be a zero of the entire function $$F(\lambda)=\sum_{j=1}^n e^{\lambda\eta_j}.$$ ...
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Asymptotic analysis of an expression involving a Fox's H function

Your expression can be simplified as follows: As $m=q=4$, the denominator parameter $(0,1)$ cancels with the numerator parameter $(0,1)$. Furthermore, as all linear coefficients are one, the Fox H ...
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The exact constant in a bound on ratios of Gamma functions

The optimal $C$ is $\mathrm{e}$. Proof: We have $$\ln C \ge \ln a - \ln(a + b) + \frac{\ln \Gamma(a + b) -\ln a - \ln\Gamma(a) - \ln\Gamma(b)}{a}.$$ Let F(a, b) := \ln a - \ln(a + b) + \frac{\ln ...
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The expression obtained by @Robert can be simplified with the Imaginary-Argument Transformation from DLMF 19.7.7. Then, the limit $t\to 1^-$ can be performed. The result can be further simplified and ...