The DerivativeRepresentation(S, x) calling sequence returns a series equal to S written in terms of the family of polynomials produced by differentiating the S polynomials with respect to x.

The DerivativeRepresentation(S, x1,.., xn) and DerivativeRepresentation(S, [x1,.., xn]) calling sequences are equivalent to the recursive calling sequence DerivativeRepresentation(...DerivativeRepresentation(S, x1),..., xn).

The partial differential representation can be used for continuous hypergeometric polynomials with a degree 2 sigma polynomial. The partial differential representation (with respect to the root xi for the polynomials poly(n, x) depending on x in the series S) is obtained by using the DerivativeRepresentation(S, x, root=val) calling sequence. If val is not a root of the sigma associated with poly(n, x), an error message is returned. The DerivativeRepresentation(S, x1,.., xn, root=val) and DerivativeRepresentation(S, [x1,.., xn], root=val) calling sequences assume that all polynomials depending on x1,.., xn share the common root val. Otherwise, an error is returned.

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

$\mathrm{S1}≔\mathrm{Create}\left(a\left(n\,m\right)\,\mathrm{LaguerreL}\left(n\,\mathrm{\alpha }\,x\right)\,\mathrm{LaguerreL}\left(m\,\mathrm{\beta }\,y\right)\right)$

$\mathrm{S1}≔\sum _{m=0}^{\mathrm{\infty }}\left(\sum _{n=0}^{\mathrm{\infty }}a\left(n\,m\right)\mathrm{LaguerreL}\left(n\,\mathrm{\alpha }\,x\right)\right)\mathrm{LaguerreL}\left(m\,\mathrm{\beta }\,y\right)$

$\mathrm{S2}≔\mathrm{DerivativeRepresentation}\left(\mathrm{S1}\,x\right)$

$\mathrm{S2}≔\sum _{m=0}^{\mathrm{\infty }}\left(\sum _{n=0}^{\mathrm{\infty }}\left(a\left(n\,m\right)-a\left(n+1\,m\right)\right)\mathrm{LaguerreL}\left(n\,\mathrm{\alpha }+1\,x\right)\right)\mathrm{LaguerreL}\left(m\,\mathrm{\beta }\,y\right)$

$\mathrm{DerivativeRepresentation}\left(\mathrm{S2}\,y\right)\;$$\mathrm{DerivativeRepresentation}\left(\mathrm{S1}\,\left[x\,y\right]\right)$

$\sum _{m=0}^{\mathrm{\infty }}\left(\sum _{n=0}^{\mathrm{\infty }}\left(a\left(n\,m\right)-a\left(n+1\,m\right)-a\left(n\,m+1\right)+a\left(n+1\,m+1\right)\right)\mathrm{LaguerreL}\left(n\,\mathrm{\alpha }+1\,x\right)\right)\mathrm{LaguerreL}\left(m\,\mathrm{\beta }+1\,y\right)$

$\sum _{m=0}^{\mathrm{\infty }}\left(\sum _{n=0}^{\mathrm{\infty }}a\left(n\,m\right)\mathrm{LaguerreL}\left(n\,\mathrm{\alpha }\,x\right)\right)\mathrm{LaguerreL}\left(m\,\mathrm{\beta }\,y\right)$

$\mathrm{S5}≔\mathrm{Create}\left(\frac{1}{n+1}\,\mathrm{JacobiP}\left(n\,1\,2\,x\right)\right)$

$\mathrm{S5}≔\sum _{n=0}^{\mathrm{\infty }}\frac{\mathrm{JacobiP}\left(n\,1\,2\,x\right)}{n+1}$

$\mathrm{DerivativeRepresentation}\left(\mathrm{S5}\,x\,\mathrm{root}=1\right)$

$\sum _{n=0}^{\mathrm{\infty }}\frac{\left(n^2+5n+7\right)\mathrm{JacobiP}\left(n\,1\,3\,x\right)}{\left(n+3\right)\left(n+2\right)\left(n+1\right)}$

$\mathrm{DerivativeRepresentation}\left(\mathrm{S5}\,x\,\mathrm{root}=-1\right)$

$\sum _{n=0}^{\mathrm{\infty }}\frac{3\mathrm{JacobiP}\left(n\,2\,2\,x\right)}{2\left(n+2\right)\left(n+1\right)}$

$\mathrm{DerivativeRepresentation}\left(\mathrm{S5}\,x\,\mathrm{root}=-2\right)$

Error, (in OrthogonalSeries:-DerivativeRepresentation) -2 is not a root of x^2-1