The Multiply(S1, S2) function multiplies the series S2 by the finite series S1. The routine uses an adaptation of the Horner scheme.

The result is expanded in terms of the polynomials of S2.

This command is part of the OrthogonalSeries package, so it can be used in the form Multiply(..) only after executing the command with(OrthogonalSeries). However, it can always be accessed through the long form of the command by using OrthogonalSeries[Multiply](..).

$\mathrm{with}\left(\mathrm{OrthogonalSeries}\right)\:$

$\mathrm{S1}≔\mathrm{ChangeBasis}\left(1+x^2\,\mathrm{LaguerreL}\left(n\,\frac{1}{3}\,x\right)\right)$

$\mathrm{S1}≔\frac{37\mathrm{LaguerreL}\left(0\,\frac{1}{3}\,x\right)}{9}-\frac{14\mathrm{LaguerreL}\left(1\,\frac{1}{3}\,x\right)}{3}+2\mathrm{LaguerreL}\left(2\,\frac{1}{3}\,x\right)$

$\mathrm{S2}≔\mathrm{Create}\left(u\left(m\right)\,\mathrm{Charlier}\left(m\,\frac{1}{7}\,x\right)\right)$

$\mathrm{S2}≔\sum _{m=0}^{\mathrm{\infty }}u\left(m\right)\mathrm{Charlier}\left(m\,\frac{1}{7}\,x\right)$

$\mathrm{Multiply}\left(\mathrm{S1}\,\mathrm{S1}\right)$

$\frac{4225\mathrm{LaguerreL}\left(0\,\frac{1}{3}\,x\right)}{81}-\frac{3892\mathrm{LaguerreL}\left(1\,\frac{1}{3}\,x\right)}{27}+\frac{532\mathrm{LaguerreL}\left(2\,\frac{1}{3}\,x\right)}{3}-104\mathrm{LaguerreL}\left(3\,\frac{1}{3}\,x\right)+24\mathrm{LaguerreL}\left(4\,\frac{1}{3}\,x\right)$

$\mathrm{SimplifyCoefficients}\left(\mathrm{Multiply}\left(\mathrm{S1}\,\mathrm{S2}\right)\,\mathrm{collect}\,u\right)$

$\left(\frac{57u\left(0\right)}{49}-\frac{9u\left(1\right)}{7}+2u\left(2\right)\right)\mathrm{Charlier}\left(0\,\frac{1}{7}\,x\right)+\left(\frac{134u\left(1\right)}{49}-\frac{9u\left(0\right)}{49}-\frac{46u\left(2\right)}{7}+6u\left(3\right)\right)\mathrm{Charlier}\left(1\,\frac{1}{7}\,x\right)+\left(\sum _{m=2}^{\mathrm{\infty }}\left(\left(\frac{57}{49}+\frac{4}{7}m+m^2\right)u\left(m\right)+\frac{u\left(m-2\right)}{49}+\left(\frac{5}{49}-\frac{2m}{7}\right)u\left(m-1\right)+\left(-\frac{23}{7}m-\frac{9}{7}-2m^2\right)u\left(m+1\right)+\left(m^2+3m+2\right)u\left(m+2\right)\right)\mathrm{Charlier}\left(m\,\frac{1}{7}\,x\right)\right)$

$\mathrm{Multiply}\left(\mathrm{S2}\,\mathrm{S2}\right)$

Error, (in OrthogonalSeries:-Multiply) The first series  must be finite