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OrthogonalSeries

 Derivate
 take derivative of a series

 Calling Sequence Derivate(S, x, option_operator, optional_root) Derivate(S, x1,.., xn, option_operator, optional_root) Derivate(S, [x1,.., xn], option_operator, optional_root)

Parameters

 S - orthogonal series x, x1, .., xn - names option_operator - (optional) equation of the form operator = operator_name, where operator_name is chosen from: deltastruct, nablastruct, hypdiff, invqstruct, qstruct, struct, and structural optional_root - (optional) equation of the form root = val where val is a symbol representing a root of the polynomial associated with the expansion family

Description

 • The Derivate(S, x) calling sequence applies to the series S the differential, difference or q-difference operator with respect to the variable x. The operator is determined by the hypergeometric type of the polynomials depending on x in the series. For example, if ChebyshevT(n, x) and Charlier(m, a, y) appear in the series S, then Derivate(S, x) differentiates S with respect to x and Derivate(S, y) applies to S the delta difference operator with respect to y.
 • The Derivate(S, x1,.., xn) and Derivate(S, [x1,.., xn]) calling sequences are equivalent to the recursive calling sequence Derivate(...Derivate(S, x1),..., xn).
 • The operator=operator_name option allows the user to apply a special difference/differential operator to a series. The Derivate(S, x1,.., xn, operator=operator_name) and Derivate(S, [x1,.., xn], operator=operator_name) calling sequences are equivalent to the recursive Derivate(...Derivate(S, x1, operator=operator_name)..., xn, operator=operator_name) calling sequence.
 • If operator_name is:
 deltastruct or nablastruct, then the discrete delta-structural or nabla-structural operator, respectively, is applied.
 hypdiff, then the hypergeometric operator is applied to the continuous, discrete, or q-discrete polynomial.
 invqstruct or qstruct, then the 1/q-structural or q-structural operator, respectively, is applied.
 struct, then the structural differential operator is applied.
 structural, then the structural differential (for continuous hypergeometric polynomials), the discrete nabla-structural (for discrete hypergeometric polynomials), or the q-structural (for q-discrete hypergeometric polynomials) operator is applied.
 • The operator type can be incompatible with the polynomial to which it is applied.  For example, the q-structural operator with respect to x cannot be applied (in general or with the OrthogonalSeries package) to a series S expanded in terms of the Hermite polynomials HermiteH(n, x). In such a case, an error message is returned.
 • This command is part of the OrthogonalSeries package, so it can be used in the form Derivate(..) only after executing the command with(OrthogonalSeries). However, it can always be accessed through the l ong form of the command by using OrthogonalSeries[Derivate](..).

Examples

 > $\mathrm{with}\left(\mathrm{OrthogonalSeries}\right):$
 > $S≔\mathrm{Create}\left(u\left(n,m\right),\mathrm{ChebyshevT}\left(n,x\right),\mathrm{GegenbauerC}\left(m,a,y\right)\right)$
 ${S}{:=}{\sum }_{{m}{=}{0}}^{{\mathrm{∞}}}\phantom{\rule[-0.0ex]{5.0px}{0.0ex}}\left({\sum }_{{n}{=}{0}}^{{\mathrm{∞}}}\phantom{\rule[-0.0ex]{5.0px}{0.0ex}}{u}{}\left({n}{,}{m}\right){}{\mathrm{ChebyshevT}}{}\left({n}{,}{x}\right)\right){}{\mathrm{GegenbauerC}}{}\left({m}{,}{a}{,}{y}\right)$ (1)
 > $\mathrm{Derivate}\left(S,\left[x,y\right]\right)$
 ${\sum }_{{m}{=}{0}}^{{\mathrm{∞}}}\phantom{\rule[-0.0ex]{5.0px}{0.0ex}}\left({\sum }_{{n}{=}{0}}^{{\mathrm{∞}}}\phantom{\rule[-0.0ex]{5.0px}{0.0ex}}{2}{}{a}{}\left({n}{+}{1}\right){}{u}{}\left({n}{+}{1}{,}{m}{+}{1}\right){}{\mathrm{ChebyshevU}}{}\left({n}{,}{x}\right)\right){}{\mathrm{GegenbauerC}}{}\left({m}{,}{a}{+}{1}{,}{y}\right)$ (2)
 > $\mathrm{S1}≔\mathrm{Create}\left(u\left(n,m\right),\mathrm{ChebyshevT}\left(n,x\right),\mathrm{GegenbauerC}\left(m,a,y\right)\right)$
 ${\mathrm{S1}}{:=}{\sum }_{{m}{=}{0}}^{{\mathrm{∞}}}\phantom{\rule[-0.0ex]{5.0px}{0.0ex}}\left({\sum }_{{n}{=}{0}}^{{\mathrm{∞}}}\phantom{\rule[-0.0ex]{5.0px}{0.0ex}}{u}{}\left({n}{,}{m}\right){}{\mathrm{ChebyshevT}}{}\left({n}{,}{x}\right)\right){}{\mathrm{GegenbauerC}}{}\left({m}{,}{a}{,}{y}\right)$ (3)
 > $\mathrm{Derivate}\left(\mathrm{S1},x\right);$$\mathrm{Derivate}\left(\mathrm{S1},y\right);$$\mathrm{Derivate}\left(\mathrm{S1},\left[x,y\right]\right)$
 ${\sum }_{{m}{=}{0}}^{{\mathrm{∞}}}\phantom{\rule[-0.0ex]{5.0px}{0.0ex}}\left({\sum }_{{n}{=}{0}}^{{\mathrm{∞}}}\phantom{\rule[-0.0ex]{5.0px}{0.0ex}}\left({n}{+}{1}\right){}{u}{}\left({n}{+}{1}{,}{m}\right){}{\mathrm{ChebyshevU}}{}\left({n}{,}{x}\right)\right){}{\mathrm{GegenbauerC}}{}\left({m}{,}{a}{,}{y}\right)$
 ${\sum }_{{m}{=}{0}}^{{\mathrm{∞}}}\phantom{\rule[-0.0ex]{5.0px}{0.0ex}}\left({\sum }_{{n}{=}{0}}^{{\mathrm{∞}}}\phantom{\rule[-0.0ex]{5.0px}{0.0ex}}{2}{}{a}{}{u}{}\left({n}{,}{m}{+}{1}\right){}{\mathrm{ChebyshevT}}{}\left({n}{,}{x}\right)\right){}{\mathrm{GegenbauerC}}{}\left({m}{,}{a}{+}{1}{,}{y}\right)$
 ${\sum }_{{m}{=}{0}}^{{\mathrm{∞}}}\phantom{\rule[-0.0ex]{5.0px}{0.0ex}}\left({\sum }_{{n}{=}{0}}^{{\mathrm{∞}}}\phantom{\rule[-0.0ex]{5.0px}{0.0ex}}{2}{}{a}{}\left({n}{+}{1}\right){}{u}{}\left({n}{+}{1}{,}{m}{+}{1}\right){}{\mathrm{ChebyshevU}}{}\left({n}{,}{x}\right)\right){}{\mathrm{GegenbauerC}}{}\left({m}{,}{a}{+}{1}{,}{y}\right)$ (4)

The structural operator for Laguerre polynomials in x with the parameter a is $\frac{xd}{\mathrm{dx}}$ and the hypergeometric operator is $\frac{x{d}^{2}}{{\mathrm{dx}}^{2}}-\frac{\left(x-a-1\right)d}{\mathrm{dx}}$. We verify these results by using the ApplyOperator and DerivateRepresentation procedures.

 > $\mathrm{S1}≔\mathrm{Create}\left(\left[1,2,3,4\right],\mathrm{LaguerreL}\left(m,a,x\right)\right)$
 ${\mathrm{S1}}{:=}{\mathrm{LaguerreL}}{}\left({0}{,}{a}{,}{x}\right){+}{2}{}{\mathrm{LaguerreL}}{}\left({1}{,}{a}{,}{x}\right){+}{3}{}{\mathrm{LaguerreL}}{}\left({2}{,}{a}{,}{x}\right){+}{4}{}{\mathrm{LaguerreL}}{}\left({3}{,}{a}{,}{x}\right)$ (5)
 > $\mathrm{S2}≔\mathrm{Derivate}\left(\mathrm{S1},x,\mathrm{operator}=\mathrm{structural}\right)$
 ${\mathrm{S2}}{:=}\left({-}{2}{}{a}{-}{2}\right){}{\mathrm{LaguerreL}}{}\left({0}{,}{a}{,}{x}\right){+}\left({-}{4}{-}{3}{}{a}\right){}{\mathrm{LaguerreL}}{}\left({1}{,}{a}{,}{x}\right){+}\left({-}{6}{-}{4}{}{a}\right){}{\mathrm{LaguerreL}}{}\left({2}{,}{a}{,}{x}\right){+}{12}{}{\mathrm{LaguerreL}}{}\left({3}{,}{a}{,}{x}\right)$ (6)
 > $\mathrm{DerivativeRepresentation}\left(\mathrm{S2},x\right);$$\mathrm{ApplyOperator}\left(\left[xd,\left[d,x\right]\right],\mathrm{S1}\right)$
 $\left({2}{+}{a}\right){}{\mathrm{LaguerreL}}{}\left({0}{,}{a}{+}{1}{,}{x}\right){+}\left({2}{+}{a}\right){}{\mathrm{LaguerreL}}{}\left({1}{,}{a}{+}{1}{,}{x}\right){+}\left({-}{18}{-}{4}{}{a}\right){}{\mathrm{LaguerreL}}{}\left({2}{,}{a}{+}{1}{,}{x}\right){+}{12}{}{\mathrm{LaguerreL}}{}\left({3}{,}{a}{+}{1}{,}{x}\right)$
 $\left({2}{+}{a}\right){}{\mathrm{LaguerreL}}{}\left({0}{,}{a}{+}{1}{,}{x}\right){+}\left({2}{+}{a}\right){}{\mathrm{LaguerreL}}{}\left({1}{,}{a}{+}{1}{,}{x}\right){+}\left({-}{18}{-}{4}{}{a}\right){}{\mathrm{LaguerreL}}{}\left({2}{,}{a}{+}{1}{,}{x}\right){+}{12}{}{\mathrm{LaguerreL}}{}\left({3}{,}{a}{+}{1}{,}{x}\right)$ (7)
 > $\mathrm{S3}≔\mathrm{Derivate}\left(\mathrm{S1},x,\mathrm{operator}=\mathrm{hypdiff}\right)$
 ${\mathrm{S3}}{:=}{-}{2}{}{\mathrm{LaguerreL}}{}\left({1}{,}{a}{,}{x}\right){-}{6}{}{\mathrm{LaguerreL}}{}\left({2}{,}{a}{,}{x}\right){-}{12}{}{\mathrm{LaguerreL}}{}\left({3}{,}{a}{,}{x}\right)$ (8)
 > $\mathrm{DerivativeRepresentation}\left(\mathrm{S3},\left[x,x\right]\right);$$\mathrm{ApplyOperator}\left(\left[x{d}^{2}-\left(x-a-1\right)d,\left[d,x\right]\right],\mathrm{S1}\right)$
 ${-}{2}{}{\mathrm{LaguerreL}}{}\left({1}{,}{a}{,}{x}\right){-}{6}{}{\mathrm{LaguerreL}}{}\left({2}{,}{a}{,}{x}\right){-}{12}{}{\mathrm{LaguerreL}}{}\left({3}{,}{a}{,}{x}\right)$
 ${-}{2}{}{\mathrm{LaguerreL}}{}\left({0}{,}{2}{+}{a}{,}{x}\right){-}{2}{}{\mathrm{LaguerreL}}{}\left({1}{,}{2}{+}{a}{,}{x}\right){+}{18}{}{\mathrm{LaguerreL}}{}\left({2}{,}{2}{+}{a}{,}{x}\right){-}{12}{}{\mathrm{LaguerreL}}{}\left({3}{,}{2}{+}{a}{,}{x}\right)$ (9)

Find the structural operators for Charlier polynomials.

 > $S≔\mathrm{Create}\left(u\left(n\right),\mathrm{Charlier}\left(n,\frac{1}{2},x\right)\right)$
 ${S}{:=}{\sum }_{{n}{=}{0}}^{{\mathrm{∞}}}\phantom{\rule[-0.0ex]{5.0px}{0.0ex}}{u}{}\left({n}\right){}{\mathrm{Charlier}}{}\left({n}{,}\frac{{1}}{{2}}{,}{x}\right)$ (10)
 > $\mathrm{Derivate}\left(S,x\right)$
 ${\sum }_{{n}{=}{0}}^{{\mathrm{∞}}}\phantom{\rule[-0.0ex]{5.0px}{0.0ex}}\left({-}{2}{}\left({n}{+}{1}\right){}{u}{}\left({n}{+}{1}\right){}{\mathrm{Charlier}}{}\left({n}{,}\frac{{1}}{{2}}{,}{x}\right)\right)$ (11)
 > $\mathrm{Derivate}\left(S,x,\mathrm{operator}=\mathrm{structural}\right)$
 ${\sum }_{{n}{=}{0}}^{{\mathrm{∞}}}\phantom{\rule[-0.0ex]{5.0px}{0.0ex}}\left({n}{}{u}{}\left({n}\right){-}{u}{}\left({n}{+}{1}\right){}{n}{-}{u}{}\left({n}{+}{1}\right)\right){}{\mathrm{Charlier}}{}\left({n}{,}\frac{{1}}{{2}}{,}{x}\right)$ (12)
 > $\mathrm{Derivate}\left(S,x,\mathrm{operator}=\mathrm{nablastruct}\right)$
 ${\sum }_{{n}{=}{0}}^{{\mathrm{∞}}}\phantom{\rule[-0.0ex]{5.0px}{0.0ex}}\left({n}{}{u}{}\left({n}\right){-}{u}{}\left({n}{+}{1}\right){}{n}{-}{u}{}\left({n}{+}{1}\right)\right){}{\mathrm{Charlier}}{}\left({n}{,}\frac{{1}}{{2}}{,}{x}\right)$ (13)
 > $\mathrm{Derivate}\left(S,x,\mathrm{operator}=\mathrm{deltastruct}\right)$
 ${\sum }_{{n}{=}{0}}^{{\mathrm{∞}}}\phantom{\rule[-0.0ex]{5.0px}{0.0ex}}\left({-}\left({n}{+}{1}\right){}{u}{}\left({n}{+}{1}\right){}{\mathrm{Charlier}}{}\left({n}{,}\frac{{1}}{{2}}{,}{x}\right)\right)$ (14)

Find the structural operators for Moak q-polynomials.

 > $S≔\mathrm{Create}\left(u\left(n\right),\mathrm{Moak}\left(n,2,\frac{1}{3},x\right)\right)$
 ${S}{:=}{\sum }_{{n}{=}{0}}^{{\mathrm{∞}}}\phantom{\rule[-0.0ex]{5.0px}{0.0ex}}{u}{}\left({n}\right){}{\mathrm{Moak}}{}\left({n}{,}{2}{,}\frac{{1}}{{3}}{,}{x}\right)$ (15)
 > $R≔\mathrm{Derivate}\left(S,x,\mathrm{operator}=\mathrm{qstruct}\right)$
 ${R}{:=}{-}{117}{}{u}{}\left({1}\right){}{\mathrm{Moak}}{}\left({0}{,}{2}{,}\frac{{1}}{{3}}{,}{x}\right){+}{\sum }_{{n}{=}{1}}^{{\mathrm{∞}}}\phantom{\rule[-0.0ex]{5.0px}{0.0ex}}\left(\frac{{1}}{{27}}{}{u}{}\left({n}{-}{1}\right){}{\left(\frac{{1}}{{3}}\right)}^{{n}{-}{1}}{-}\frac{{1}}{{27}}{}{u}{}\left({n}{-}{1}\right){-}\frac{{3}}{{2}}{}{\left(\frac{{1}}{{3}}\right)}^{{-}{2}{}{n}}{}{u}{}\left({n}\right){}{\left(\frac{{1}}{{3}}\right)}^{{n}}{}{\left(\frac{{1}}{{3}}\right)}^{{n}{+}{3}}{+}{2}{}{\left(\frac{{1}}{{3}}\right)}^{{-}{2}{}{n}}{}{u}{}\left({n}\right){}{\left(\frac{{1}}{{3}}\right)}^{{n}}{+}\frac{{3}}{{2}}{}{\left(\frac{{1}}{{3}}\right)}^{{-}{2}{}{n}}{}{u}{}\left({n}\right){}{\left(\frac{{1}}{{3}}\right)}^{{n}{+}{3}}{-}{2}{}{\left(\frac{{1}}{{3}}\right)}^{{-}{2}{}{n}}{}{u}{}\left({n}\right){-}\frac{{9}}{{4}}{}{\left(\frac{{1}}{{3}}\right)}^{{n}{+}{3}}{}{\left(\frac{{1}}{{3}}\right)}^{{-}{4}{-}{4}{}{n}}{}{\left(\frac{{1}}{{3}}\right)}^{{n}{+}{1}}{}{u}{}\left({n}{+}{1}\right){+}\frac{{9}}{{4}}{}{\left(\frac{{1}}{{3}}\right)}^{{n}{+}{3}}{}{\left(\frac{{1}}{{3}}\right)}^{{-}{4}{-}{4}{}{n}}{}{u}{}\left({n}{+}{1}\right){+}\frac{{9}}{{4}}{}{\left(\frac{{1}}{{3}}\right)}^{{-}{4}{-}{4}{}{n}}{}{\left(\frac{{1}}{{3}}\right)}^{{n}{+}{1}}{}{u}{}\left({n}{+}{1}\right){-}\frac{{9}}{{4}}{}{\left(\frac{{1}}{{3}}\right)}^{{-}{4}{-}{4}{}{n}}{}{u}{}\left({n}{+}{1}\right)\right){}{\mathrm{Moak}}{}\left({n}{,}{2}{,}\frac{{1}}{{3}}{,}{x}\right)$ (16)
 > $\mathrm{SimplifyCoefficients}\left(R,\mathrm{collect},u,\mathrm{simplify}\right)$
 ${-}{117}{}{u}{}\left({1}\right){}{\mathrm{Moak}}{}\left({0}{,}{2}{,}\frac{{1}}{{3}}{,}{x}\right){+}{\sum }_{{n}{=}{1}}^{{\mathrm{∞}}}\phantom{\rule[-0.0ex]{5.0px}{0.0ex}}\left(\left({-}\frac{{1}}{{18}}{+}\frac{{37}}{{18}}{}{{3}}^{{n}}{-}{2}{}{{9}}^{{n}}\right){}{u}{}\left({n}\right){+}\left(\frac{{1}}{{9}}{}{{3}}^{{-}{n}}{-}\frac{{1}}{{27}}\right){}{u}{}\left({n}{-}{1}\right){+}\left({-}\frac{{1}}{{4}}{}{{3}}^{{2}{}{n}{+}{2}}{+}\frac{{135}}{{2}}{}{{27}}^{{n}}{-}\frac{{729}}{{4}}{}{{81}}^{{n}}\right){}{u}{}\left({n}{+}{1}\right)\right){}{\mathrm{Moak}}{}\left({n}{,}{2}{,}\frac{{1}}{{3}}{,}{x}\right)$ (17)

Find the partial structural operators for Jacobi polynomials.

 > $S≔\mathrm{Create}\left(u\left(n\right),\mathrm{JacobiP}\left(n,\frac{1}{2},\frac{1}{3},x\right)\right)$
 ${S}{:=}{\sum }_{{n}{=}{0}}^{{\mathrm{∞}}}\phantom{\rule[-0.0ex]{5.0px}{0.0ex}}{u}{}\left({n}\right){}{\mathrm{JacobiP}}{}\left({n}{,}\frac{{1}}{{2}}{,}\frac{{1}}{{3}}{,}{x}\right)$ (18)
 > $\mathrm{SimplifyCoefficients}\left(\mathrm{Derivate}\left(S,x,\mathrm{operator}=\mathrm{struct},\mathrm{root}=1\right),\mathrm{collect},u\right)$
 ${\sum }_{{n}{=}{0}}^{{\mathrm{∞}}}\phantom{\rule[-0.0ex]{5.0px}{0.0ex}}\left(\frac{{1}}{{2}}{}\frac{\left({144}{}{{n}}^{{3}}{+}{540}{}{{n}}^{{2}}{+}{506}{}{n}\right){}{u}{}\left({n}\right)}{\left({12}{}{n}{+}{23}\right){}\left({12}{}{n}{+}{11}\right)}{+}\frac{{1}}{{2}}{}\frac{\left({-}{144}{}{{n}}^{{3}}{-}{756}{}{{n}}^{{2}}{-}{1184}{}{n}{-}{561}\right){}{u}{}\left({n}{+}{1}\right)}{\left({12}{}{n}{+}{23}\right){}\left({12}{}{n}{+}{11}\right)}\right){}{\mathrm{JacobiP}}{}\left({n}{,}\frac{{1}}{{2}}{,}\frac{{4}}{{3}}{,}{x}\right)$ (19)
 > $\mathrm{SimplifyCoefficients}\left(\mathrm{Derivate}\left(S,x,\mathrm{operator}=\mathrm{struct},\mathrm{root}=-1\right),\mathrm{collect},u\right)$
 ${\sum }_{{n}{=}{0}}^{{\mathrm{∞}}}\phantom{\rule[-0.0ex]{5.0px}{0.0ex}}\left(\frac{{1}}{{3}}{}\frac{\left({216}{}{{n}}^{{3}}{+}{810}{}{{n}}^{{2}}{+}{759}{}{n}\right){}{u}{}\left({n}\right)}{\left({12}{}{n}{+}{23}\right){}\left({12}{}{n}{+}{11}\right)}{+}\frac{{1}}{{3}}{}\frac{\left({216}{}{{n}}^{{3}}{+}{1098}{}{{n}}^{{2}}{+}{1641}{}{n}{+}{748}\right){}{u}{}\left({n}{+}{1}\right)}{\left({12}{}{n}{+}{23}\right){}\left({12}{}{n}{+}{11}\right)}\right){}{\mathrm{JacobiP}}{}\left({n}{,}\frac{{3}}{{2}}{,}\frac{{1}}{{3}}{,}{x}\right)$ (20)