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The Maxima on-line user's manual

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Hermite

Function: hermite (<n>, <x>) The Hermite polynomial.

sum(t^n/n! * diff(diff(%e^(x*y*t), x, n), y, n), n, 0, inf), simpsum=true;
sum(t^n/n! * t^n * hermite(n, t*x*y), n, 0, inf), simpsum=true;
diff(diff(%e^(x*y*t), x, 1), y, 1);
diff(diff(%e^(x*y*t), x, 2), y, 2);
diff(diff(%e^(x*y*t), x, 3), y, 3);

Reference: Abramowitz and Stegun, equation 22.5.55, page 780.

(%o1)                                true
(%i2) 

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