Closed Form Evaluation of

This is the Mehler formula for Hermite polynomials. We first input the summand.

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Since the sum is over , we need the dependency of the summand in terms of . In this session, we focus on the dependency of the sum in , and leave and as parameters, but other choices could have been made. The following call yields a system that traces the dependency of the summand in and .

> sys[1]:=dfinite_expr_to_sys(expr,f(n::shift,u::diff));

The summation itself is performed by the following call. Viewing the sum as a formal power series in , we have that the summand goes to 0 as tends to infinity. At the other boundary, the recurrence in proves that the summand is 0 for negative . The use of the option is therefore fully justified.

> sys[2]:=sum_of_sys(sys[1],n=0..infinity,natural_boundaries);

There only remains to solve this differential equation.

> dsolve(sys[2] union {f(0)=1},f(u));

> simplify(expand(%),symbolic);

In other words, we have obtained: .