Let $t_1,\ldots,t_m>0$, and $m\ge 4$ be an even integer. Consider the function: $$ f(a,b;\mathbf{t})=\int_0^{t_1}\ldots\int_0^{t_m} |x_1-x_m|^a |x_2-x_1|^b |x_3-x_2|^a |x_4-x_3|^b \ldots |x_{m-1}-x_{m-2}|^a |x_{m}-x_{m-1}|^b d\mathbf{x}. $$ (It is finite when $a,b>-1$ and $a+b>2(1/m-1)$. See 1.)
The goal is to study the asymptotics of $f(a,b;\mathbf{t})$ as $(a,b)\rightarrow(0,-1)$.
Heuristically, by first setting $a=0$, and then using the identity (for $b>-1$): $$ \int_0^{t_1}\int_0^{t_2}|x_1-x_2|^{b} dx_1dx_2=\frac{1}{(b+1)(b+2)} \left( t_1^{b+2}+ t_2^{b+2}-|t_1-t_2|^{b+2} \right), $$ one expects as $(a,b)\rightarrow(0,-1)$ that $$ f(a,b;\mathbf{t})\sim (b+1)^{-m/2} \prod_{i=2,4,...m} \left( t_i+ t_{i-1}-|t_{i}-t_{i-1}| \right). $$ But I had a hard time justifying this (if it is correct) rigorously. The Dominated Convergence seems not directly applicable because as $b\rightarrow -1$ the integrand is not bounded by an integrable function.
Could any one offer some idea? Many thanks.