I already asked this a few weeks ago with no answer, so let me formulate differently.

In performing Lagrange interpolation with nodes 1/n, one encounters the sum $$S(f)=\dfrac{1}{N!}\sum_{n=0}^N(-1)^{N-n}\binom{N}{n}n^Nf(n)\;.$$

1. If $f(n)=n^{-a}$ for any real (or complex ?) $a$ it seems that $S(f)$ is asymptotic as $N\to\infty$ to $$\dfrac{2^a}{\Gamma(1-a)}N^{-2a}$$ (including when $a$ is a positive or $0$ integer). This is probably easy but I haven't found the result, certainly in the literature.

2. My real question is what is the asymptotic when $f(n)=\psi'(n)$, where $\psi$ is the logarithmic derivative of the gamma function. I believe that up to smaller order terms it is $C^{-N}$, but what is $C$ ? I also believe that we have the same asymptotic with the same $C$ for higher derivatives of $\psi$.

ADDED for Q 2): I had a heuristic which gave $C=2\pi$, but the correct value seems to be $C=8.077...$.

I already asked this a few weeks ago with no answer, so let me formulate differently.

In performing Lagrange interpolation with nodes 1/n, one encounters the sum $$S(f)=\dfrac{1}{N!}\sum_{n=0}^N(-1)^{N-n}\binom{N}{n}n^Nf(n)\;.$$

1. If $f(n)=n^{-a}$ for any real (or complex ?) $a$ it seems that $S(f)$ is asymptotic as $N\to\infty$ to $$\dfrac{2^a}{\Gamma(1-a)}N^{-2a}$$ (including when $a$ is a positive or $0$ integer). This is probably easy but I haven't found the result, certainly in the literature.

2. My real question is what is the asymptotic when $f(n)=\psi'(n)$, where $\psi$ is the logarithmic derivative of the gamma function. I believe that up to smaller order terms it is $C^{-N}$, but what is $C$ ? I also believe that we have the same asymptotic with the same $C$ for higher derivatives of $\psi$.

ADDED for Q 2): I had a heuristic which gave $C=2\pi$, but the correct value seems to be $C=8.077...$.

UPDATE!!!: Don Zagier has completely answered Q 1), and almost completely Q 2: the constant $C$ is in fact a complex number $x_0$ solution of $\exp(x_0-1)=x_0$, whose modulus is indeed approximately $8.076$

I already asked this a few weeks ago with no answer, so let me formulate differently.

In performing Lagrange interpolation with nodes 1/n, one encounters the sum $$S(f)=\dfrac{1}{N!}\sum_{n=0}^N(-1)^{N-n}\binom{N}{n}n^Nf(n)\;.$$

1. If $f(n)=n^{-a}$ for any real (or complex ?) $a$ it seems that $S(f)$ is asymptotic as $N\to\infty$ to $$\dfrac{2^a}{\Gamma(1-a)}N^{-2a}$$ (including when $a$ is a positive or $0$ integer). This is probably easy but I haven't found the result, certainly in the literature.

2. My real question is what is the asymptotic when $f(n)=\psi'(n)$, where $\psi$ is the logarithmic derivative of the gamma function. I believe that up to smaller order terms it is $C^{-N}$, but what is $C$ ? I also believe that we have the same asymptotic with the same $C$ for higher derivatives of $\psi$.

I already asked this a few weeks ago with no answer, so let me formulate differently.

In performing Lagrange interpolation with nodes 1/n, one encounters the sum $$S(f)=\dfrac{1}{N!}\sum_{n=0}^N(-1)^{N-n}\binom{N}{n}n^Nf(n)\;.$$

1. If $f(n)=n^{-a}$ for any real (or complex ?) $a$ it seems that $S(f)$ is asymptotic as $N\to\infty$ to $$\dfrac{2^a}{\Gamma(1-a)}N^{-2a}$$ (including when $a$ is a positive or $0$ integer). This is probably easy but I haven't found the result, certainly in the literature.

2. My real question is what is the asymptotic when $f(n)=\psi'(n)$, where $\psi$ is the logarithmic derivative of the gamma function. I believe that up to smaller order terms it is $C^{-N}$, but what is $C$ ? I also believe that we have the same asymptotic with the same $C$ for higher derivatives of $\psi$.

ADDED for Q 2): I had a heuristic which gave $C=2\pi$, but the correct value seems to be $C=8.077...$.