OrthogonalSeries - Maple Programming Help

OrthogonalSeries

 add or form linear combination of series

 Calling Sequence Add(S1, S2, a, b, optional_arg)

Parameters

 S1, S2 - orthogonal series a, b - (optional) scalars optional_arg - (optional) equation of the form control = true or false

Description

 • The Add(S1, S2) calling sequence adds the series S1 and S2.
 • The Add(S1, S2, a) and Add(S1, S2, a, b) calling sequences form the linear combinations $\mathrm{S1}a+\mathrm{S2}$ and $\mathrm{S1}a+\mathrm{S2}b$, respectively.
 • S1 and S2 must have the same dimension. Each polynomial appearing in S1 must appear in S2 with the same variable(s). Otherwise, an error message is returned.  The indices of S1 and S2 may differ.  The indices of the result are those of S1.
 • The default value of the control option is true. If the option control=false is used, the Add function assumes the polynomials of S2 are identical to those of S1. This allows faster addition if the series are known a priori to be compatible. However, if S1 and S2 are not compatible, it returns an incorrect result.

Examples

 > $\mathrm{with}\left(\mathrm{OrthogonalSeries}\right):$
 > $\mathrm{S1}≔\mathrm{Create}\left(n+1,\mathrm{Kravchouk}\left(n,p,q,x\right)\right)$
 ${\mathrm{S1}}{≔}{\sum }_{{n}{=}{0}}^{{\mathrm{∞}}}\phantom{\rule[-0.0ex]{5.0px}{0.0ex}}\left({n}{+}{1}\right){}{\mathrm{Kravchouk}}{}\left({n}{,}{p}{,}{q}{,}{x}\right)$ (1)
 > $\mathrm{S2}≔\mathrm{Create}\left(\frac{1}{m},\mathrm{Kravchouk}\left(m,p,q,x\right)\right)$
 ${\mathrm{S2}}{≔}{\sum }_{{m}{=}{0}}^{{\mathrm{∞}}}\phantom{\rule[-0.0ex]{5.0px}{0.0ex}}\frac{{\mathrm{Kravchouk}}{}\left({m}{,}{p}{,}{q}{,}{x}\right)}{{m}}$ (2)
 > $\mathrm{Add}\left(\mathrm{S1},\mathrm{S2}\right);$$\mathrm{Add}\left(\mathrm{S1},\mathrm{S2},a\right);$$\mathrm{Add}\left(\mathrm{S1},\mathrm{S2},a,b\right)$
 ${\sum }_{{n}{=}{0}}^{{\mathrm{∞}}}\phantom{\rule[-0.0ex]{5.0px}{0.0ex}}\left({n}{+}{1}{+}\frac{{1}}{{n}}\right){}{\mathrm{Kravchouk}}{}\left({n}{,}{p}{,}{q}{,}{x}\right)$
 ${\sum }_{{n}{=}{0}}^{{\mathrm{∞}}}\phantom{\rule[-0.0ex]{5.0px}{0.0ex}}\left({a}{}\left({n}{+}{1}\right){+}\frac{{1}}{{n}}\right){}{\mathrm{Kravchouk}}{}\left({n}{,}{p}{,}{q}{,}{x}\right)$
 ${\sum }_{{n}{=}{0}}^{{\mathrm{∞}}}\phantom{\rule[-0.0ex]{5.0px}{0.0ex}}\left({a}{}\left({n}{+}{1}\right){+}\frac{{b}}{{n}}\right){}{\mathrm{Kravchouk}}{}\left({n}{,}{p}{,}{q}{,}{x}\right)$ (3)
 > $\mathrm{S3}≔\mathrm{Create}\left(\frac{1}{n!},\mathrm{ChebyshevT}\left(n,x\right)\right)$
 ${\mathrm{S3}}{≔}{\sum }_{{n}{=}{0}}^{{\mathrm{∞}}}\phantom{\rule[-0.0ex]{5.0px}{0.0ex}}\frac{{\mathrm{ChebyshevT}}{}\left({n}{,}{x}\right)}{{n}{!}}$ (4)
 > $\mathrm{Add}\left(\mathrm{S1},\mathrm{S3}\right);$$\mathrm{Add}\left(\mathrm{S1},\mathrm{S3},\mathrm{control}=\mathrm{false}\right)$
 ${\sum }_{{n}{=}{0}}^{{\mathrm{∞}}}\phantom{\rule[-0.0ex]{5.0px}{0.0ex}}\left({n}{+}{1}{+}\frac{{1}}{{n}{!}}\right){}{\mathrm{Kravchouk}}{}\left({n}{,}{p}{,}{q}{,}{x}\right)$ (5)
 > $\mathrm{S1}≔\mathrm{Create}\left(\left\{\frac{1}{n!},n=2..8\right\},\mathrm{ChebyshevT}\left(n,x\right)\right)$
 ${\mathrm{S1}}{≔}{\sum }_{{n}{=}{2}}^{{8}}\phantom{\rule[-0.0ex]{5.0px}{0.0ex}}\frac{{\mathrm{ChebyshevT}}{}\left({n}{,}{x}\right)}{{n}{!}}$ (6)
 > $\mathrm{S2}≔\mathrm{Create}\left(\left\{\frac{1}{n!},n=4..6\right\},\mathrm{ChebyshevT}\left(n,x\right)\right)$
 ${\mathrm{S2}}{≔}{\sum }_{{n}{=}{4}}^{{6}}\phantom{\rule[-0.0ex]{5.0px}{0.0ex}}\frac{{\mathrm{ChebyshevT}}{}\left({n}{,}{x}\right)}{{n}{!}}$ (7)
 > $\mathrm{Add}\left(\mathrm{S1},\mathrm{S2}\right)$
 $\frac{{1}}{{2}}{}{\mathrm{ChebyshevT}}{}\left({2}{,}{x}\right){+}\frac{{1}}{{6}}{}{\mathrm{ChebyshevT}}{}\left({3}{,}{x}\right){+}\frac{{1}}{{5040}}{}{\mathrm{ChebyshevT}}{}\left({7}{,}{x}\right){+}\frac{{1}}{{40320}}{}{\mathrm{ChebyshevT}}{}\left({8}{,}{x}\right){+}{\sum }_{{n}{=}{4}}^{{6}}\phantom{\rule[-0.0ex]{5.0px}{0.0ex}}\frac{{2}{}{\mathrm{ChebyshevT}}{}\left({n}{,}{x}\right)}{{n}{!}}$ (8)
 > $\mathrm{S4}≔\mathrm{Create}\left(\left\{\frac{1}{n!},n=2..\mathrm{∞}\right\},\mathrm{ChebyshevT}\left(n,x\right)\right)$
 ${\mathrm{S4}}{≔}{\sum }_{{n}{=}{2}}^{{\mathrm{∞}}}\phantom{\rule[-0.0ex]{5.0px}{0.0ex}}\frac{{\mathrm{ChebyshevT}}{}\left({n}{,}{x}\right)}{{n}{!}}$ (9)
 > $\mathrm{S5}≔\mathrm{Create}\left(\left\{m!,m=4..\mathrm{∞}\right\},\mathrm{ChebyshevT}\left(m,x\right)\right)$
 ${\mathrm{S5}}{≔}{\sum }_{{m}{=}{4}}^{{\mathrm{∞}}}\phantom{\rule[-0.0ex]{5.0px}{0.0ex}}{m}{!}{}{\mathrm{ChebyshevT}}{}\left({m}{,}{x}\right)$ (10)
 > $\mathrm{Add}\left(\mathrm{S4},\mathrm{S5}\right)$
 $\frac{{1}}{{2}}{}{\mathrm{ChebyshevT}}{}\left({2}{,}{x}\right){+}\frac{{1}}{{6}}{}{\mathrm{ChebyshevT}}{}\left({3}{,}{x}\right){+}{\sum }_{{n}{=}{4}}^{{\mathrm{∞}}}\phantom{\rule[-0.0ex]{5.0px}{0.0ex}}\left(\frac{{1}}{{n}{!}}{+}{n}{!}\right){}{\mathrm{ChebyshevT}}{}\left({n}{,}{x}\right)$ (11)
 > $\mathrm{S6}≔\mathrm{Create}\left(\left\{u\left(n,m\right)\right\},\mathrm{ChebyshevT}\left(n,x\right),\mathrm{LaguerreL}\left(m,1,y\right)\right)$
 ${\mathrm{S6}}{≔}{\sum }_{{m}{=}{0}}^{{\mathrm{∞}}}\phantom{\rule[-0.0ex]{5.0px}{0.0ex}}\left({\sum }_{{n}{=}{0}}^{{\mathrm{∞}}}\phantom{\rule[-0.0ex]{5.0px}{0.0ex}}{u}{}\left({n}{,}{m}\right){}{\mathrm{ChebyshevT}}{}\left({n}{,}{x}\right)\right){}{\mathrm{LaguerreL}}{}\left({m}{,}{1}{,}{y}\right)$ (12)
 > $\mathrm{S7}≔\mathrm{Create}\left(\left\{v\left(n,m\right)\right\},\mathrm{ChebyshevT}\left(m,x\right),\mathrm{LaguerreL}\left(n,1,y\right)\right)$
 ${\mathrm{S7}}{≔}{\sum }_{{n}{=}{0}}^{{\mathrm{∞}}}\phantom{\rule[-0.0ex]{5.0px}{0.0ex}}\left({\sum }_{{m}{=}{0}}^{{\mathrm{∞}}}\phantom{\rule[-0.0ex]{5.0px}{0.0ex}}{v}{}\left({n}{,}{m}\right){}{\mathrm{ChebyshevT}}{}\left({m}{,}{x}\right)\right){}{\mathrm{LaguerreL}}{}\left({n}{,}{1}{,}{y}\right)$ (13)
 > $\mathrm{Add}\left(\mathrm{S6},\mathrm{S7}\right)$
 ${\sum }_{{m}{=}{0}}^{{\mathrm{∞}}}\phantom{\rule[-0.0ex]{5.0px}{0.0ex}}\left({\sum }_{{n}{=}{0}}^{{\mathrm{∞}}}\phantom{\rule[-0.0ex]{5.0px}{0.0ex}}\left({u}{}\left({n}{,}{m}\right){+}{v}{}\left({m}{,}{n}\right)\right){}{\mathrm{ChebyshevT}}{}\left({n}{,}{x}\right)\right){}{\mathrm{LaguerreL}}{}\left({m}{,}{1}{,}{y}\right)$ (14)