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Register a new userA Conjectured Integer Sequence Arising From the Exponential Integral
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Let $f_0(z) = exp(z/(1-z))$, $f_1(z) = exp(1/(1-z))E_1(1/(1-z))$, where $E_1(x) = int_x^infty e^{-t}t^{-1}{,d}t$. Let $a_n = [z^n]f_0(z)$ and $b_n = [z^n]f_1(z)$ be the corresponding Maclaurin series coefficients. We show that $a_n$ and $b_n$ may be expressed in terms of confluent hypergeometric functions. We consider the asymptotic behaviour of the sequences $(a_n)$ and $(b_n)$ as $n to infty$, showing that they are closely related, and proving a conjecture of Bruno Salvy regarding $(b_n)$. Let $rho_n = a_n b_n$, so $sum rho_n z^n = (f_0,odot f_1)(z)$ is a Hadamard product. We obtain an asymptotic expansion $2n^{3/2}rho_n sim -sum d_k n^{-k}$ as $n to infty$, where the $d_kinmathbb Q$, $d_0=1$. We conjecture that $2^{6k}d_k in mathbb Z$. This has been verified for $k le 1000$.
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