Let $\psi_0 (x,t)=\frac{te^{xt}}{1-e^{-t}}$. Then $$\psi_0(0,t)=\frac{t}{1-e^{-t}};$$ $$\psi_0(x,t)=1+\sum_{n=1}^\infty \frac{t^n}{n!} B_n(x)$$ where $B_n$ is a monic polynomial of degree $n.$
Now let $$\psi_1(y,t)=\int ( \frac{1}{(1-e^{-t})^2 (y-1)} (y+(1-y)e^{-t})\log (y+(1-y)e^{-t})-\frac{1}{1-e^{-t}})dt$$ normalized so that $\Psi_1(y,0)=1.$ Then $$\psi_1(0,t)=\frac{t}{1-e^{-t}};$$ $$\psi_1(x,t)=1+\sum_{n=1}^\infty \frac{(-1)^n}{n^2(n+1)}{t^{2n}} P_n(x)$$ where $P_n$ is a monic polynomial of degree $n.$
Also, let $$\psi_2(y,t)=\int \frac{1}{\sinh(\frac{t}{2})}(1-\frac{\tanh^{-1}((y+1)\tanh(\frac{t}{2}))}{(y+1)\tanh(\frac{t}{2})})dt$$ Then $$\psi_2(0,t)=\frac{t}{1-e^{-t}};$$ $$\psi_2(x,t)=1-\sum_{n=1}^\infty \frac{t^{2n}}{2^{2n+1}n(2n+1)} Q_{2n}(x)$$ where $Q_{2n}$ is a monic polynomial of degree $2n,$ even with respect to $y+1.$
Questions. Are these polynomials known? Do they appear in any proof of index/Riemann-Roch theorems? Perhaps in relation to gamma functions for associators?
