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Trying to answer this question related to the calculation of $$I_k=\int_1^\infty x^ke^{-x}\ln(x+a)\,dx$$ assuming that $k$ is a positive integer, the following polynomials appear $$\left( \begin{array}{cc} k & P_k(a) \\ 1 & a-1 \\ 2 & a^2-2 a+2 \\ 3 & a^3-3 a^2+6 a-6 \\ 4 & a^4-4 a^3+12 a^2-24 a+24 \\ 5 & a^5-5 a^4+20 a^3-60 a^2+120 a-120 \\ 6 & a^6-6 a^5+30 a^4-120 a^3+360 a^2-720 a+720 \\ 7 & a^7-7 a^6+42 a^5-210 a^4+840 a^3-2520 a^2+5040 a-5040 \\ \end{array} \right)$$

as well as these $$\left( \begin{array}{cc} k & Q_k(a) \\ 1 & 1 \\ 2 & a-4 \\ 3 & a^2-5 a+17 \\ 4 & a^3-6 a^2+25 a-84 \\ 5 & a^4-7 a^3+35 a^2-141 a+485 \\ 6 & a^5-8 a^4+47 a^3-226 a^2+911 a-3236 \\ 7 & a^6-9 a^5+61 a^4-345 a^3+1647 a^2-6703 a+24609 \\ \end{array} \right)$$

"Simple" patterns seem to appear but what could they be ?

Claude Leibovici
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1 Answers1

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The absolute values of the coefficients of the $P_k$ can be found in The On-Line Encyclopedia of Integer Sequences® as A008279:

Triangle $T(n,k) = n!/(n-k)!$ $(0 \le k \le n)$ read by rows, giving number of permutations of $n$ things $k$ at a time.

Also called permutation coefficients.

and we get $$ P_k(a) = \sum_{j=0}^k (-1)^{k-j} \frac{k!}{j!} a^j \, . $$ These polynomials occur when integrating $x^n e^x$, see for example How to integrate $ \int x^n e^x dx$?: $$ \int {x^n e^x dx} = \bigg[\sum\limits_{k = 0}^n {( - 1)^{n - k} \frac{{n!}}{{k!}}x^k } \bigg]e^x + C, $$ so that $$ P_k(a) = e^{-a} \int_{-\infty}^a x^k e^x \, dx \, . $$

Martin R
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