The dAlembertian_series_sol command returns one formal power series solution or a set of formal power series solutions with d'Alembertian coefficients for the given linear ordinary differential equation with polynomial coefficients.

If ode is an expression, then it is equated to zero.

The command returns an error message if the differential equation ode does not satisfy the following conditions.

ode must be linear in var

ode must be homogeneous or have a right-hand side that is rational or a "nice" power series in $x$

The coefficients of ode must be polynomial in the independent variable of var, for example, $x$, over the rational number field which can be extended by one or more parameters.

A homogeneous linear ordinary differential equation with coefficients that are polynomials in $x$ has a linear space of formal power series solutions $\sum _{n=0}^{\mathrm{\infty }}v\left(n\right)P_n\left(x\right)$ where $P_n\left(x\right)$ is one of ${\left(x-a\right)}^n$, $\frac{{\left(x-a\right)}^n}{n!}$, $\frac{1}{x^n}$, or $\frac{1}{x^{n}n!}$, $a$ is the expansion point, and the sequence $v\left(n\right)$ satisfies a homogeneous linear recurrence. In the case of an inhomogeneous equation with a right-hand side that is a "nice" power series, $v\left(n\right)$ satisfies an inhomogeneous linear recurrence.

The command selects such formal power series solutions where $v\left(n\right)$ is a d'Alembertian sequence, that is, $v\left(n\right)$ is annihilated by a linear recurrence operator that can be written as a composition of first-order operators (see LinearOperators).

The command determines an integer $N\ge 0$ such that $v\left(n\right)$ can be represented in the form of a d'Alembertian term:

for all $n\ge N$, where $h_i\left(n\right)$, $1\le i\le s\+1$, is a hypergeometric term (see SumTools[Hypergeometric]):

such that $R\left(k\right)=\frac{h_i\left(k+1\right)}{h_i\left(k\right)}$ is rational in $k$ for all $k\ge N$.

x=a or 'point'=a

Specifies the expansion point in the case of a homogeneous equation or an inhomogeneous equation with rational right-hand side. It can be an algebraic number, depending rationally on some parameters, or $\mathrm{\infty }$. In the case of a "nice" series right-hand side the expansion point is given by the right-hand side and cannot be changed. If the point is given, then the command returns one formal power series solution at a with d'Alembertian coefficients if it exists; otherwise, it returns NULL. If $a$  is not given, it returns a set of formal power series solutions with d'Alembertian coefficients for all singular points of ode as well as one generic ordinary point.

'free'=C

Specifies a base name C to use for free variables C[0], C[1], etc. The default is the global name  _C. Note that the number of free variables may be less than the order of the given equation.

'indices'=[n,k]

Specifies base names for dummy variables. The default values are the global names _n and _k, respectively. The name n is used as the summation index in the power series. the names n1, n2, etc., are used as summation indices in ( + ). The name k is used as the product index in ( ++ ).

'outputHGT'=name

Specifies the form of representation of hypergeometric terms.  The default value is 'inert'.

'inert' - the hypergeometric term ( ++ ) is represented by an inert product, except for $\prod _{k=N}^{n-1}1$, which is simplified to $1$.

'rcf1' or 'rcf2' - the hypergeometric term is represented in the first or second minimal representation, respectively (see ConjugateRTerm).

'active' - the hypergeometric term is represented by non-inert products which, if possible, are computed (see product).

'outputDAT'=name

Specifies the form of representation of the sums in ( + ). The default is 'inert'.

'inert' - the sums are in the inert form, except for trivial sums of the form $\sum _{k=u}^{v-1}1$, which are simplified to $v-u$.

'gosper' - Gosper's algorithm (see Gosper) is used to find a closed form for the sums in ( + ), if possible, starting with the innermost one.

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

$\mathrm{ode}≔\left(x^2+x-2\right)\mathrm{diff}\left(y\left(x\right)\,x\,x\right)+\left(x^2-x\right)\mathrm{diff}\left(y\left(x\right)\,x\right)-\left(6x^2+7x\right)y\left(x\right)$

$\mathrm{ode}≔\left(x^2+x-2\right)\left(\frac{{\ⅆ}^2}{\ⅆx^2}y\left(x\right)\right)+\left(x^2-x\right)\left(\frac{\ⅆ}{\ⅆx}y\left(x\right)\right)-\left(6x^2+7x\right)y\left(x\right)$

$\mathrm{dAlembertian\_series\_sol}\left(\mathrm{ode}\,y\left(x\right)\,'\mathrm{outputHGT}'='\mathrm{active}'\,'\mathrm{indices}'=\left[n\,k\right]\right)$

$\left\{\frac{{\mathrm{\_C}}_0\left(\sum _{n=1}^{\mathrm{\infty }}\frac{2^n{\left(x-1\right)}^n}{\mathrm{\Gamma }\left(n\right)}\right)}{2}\,\left(\frac{315\left(\sum _{n=7}^{\mathrm{\infty }}\frac{2^n\left(n-6\right){\left(x+2\right)}^n}{\mathrm{\Gamma }\left(n+1\right)}\right)}{8}-\frac{4095}{4}-\frac{1575x}{4}-315{\left(x+2\right)}^2-\frac{315{\left(x+2\right)}^3}{2}-\frac{105{\left(x+2\right)}^4}{2}-\frac{21{\left(x+2\right)}^5}{2}\right){\mathrm{\_C}}_0+\left(-17143\left(\sum _{n=7}^{\mathrm{\infty }}\frac{2^n\left(n-6\right)\left(\sum _{\mathrm{n1}=7}^{n-1}\frac{{\left(\frac{1}{6}\right)}^{\mathrm{n1}}\mathrm{\Gamma }\left(\mathrm{n1}+1\right)}{\left(\mathrm{n1}-5\right)\left(\mathrm{n1}-6\right)}\right){\left(x+2\right)}^n}{\mathrm{\Gamma }\left(n+1\right)}\right)-\frac{40\left(\sum _{n=7}^{\mathrm{\infty }}\frac{2^n\left(n-6\right)\left(\sum _{\mathrm{n1}=7}^{n-1}\frac{{\left(\frac{1}{6}\right)}^{\mathrm{n1}}\mathrm{\Gamma }\left(\mathrm{n1}+1\right)\left(\sum _{\mathrm{n2}=7}^{\mathrm{n1}-1}\frac{\left(\mathrm{n2}-5\right){\left(\mathrm{-9}\right)}^{\mathrm{n2}}}{\mathrm{\Gamma }\left(\mathrm{n2}\right)}\right)}{\left(\mathrm{n1}-5\right)\left(\mathrm{n1}-6\right)}\right){\left(x+2\right)}^n}{\mathrm{\Gamma }\left(n+1\right)}\right)}{9}+\frac{997295}{486}+\frac{383575x}{486}+\frac{153430{\left(x+2\right)}^2}{243}+\frac{77435{\left(x+2\right)}^3}{243}+\frac{73295{\left(x+2\right)}^4}{729}+\frac{18835{\left(x+2\right)}^5}{729}-\frac{2540{\left(x+2\right)}^6}{729}\right){\mathrm{\_C}}_1\,\left(\frac{\left(\sum _{\mathrm{\_n}=9}^{\mathrm{\infty }}\frac{\left(\mathrm{\_n}-8\right)2^{\mathrm{\_n}}{\left(x+3\right)}^{\mathrm{\_n}}}{\mathrm{\_n}!}\right)}{512}-\frac{25}{256}-\frac{7x}{256}-\frac{3{\left(x+3\right)}^2}{128}-\frac{5{\left(x+3\right)}^3}{384}-\frac{{\left(x+3\right)}^4}{192}-\frac{{\left(x+3\right)}^5}{640}-\frac{{\left(x+3\right)}^6}{2880}-\frac{{\left(x+3\right)}^7}{20160}\right){\mathrm{\_C}}_0+\left(-\frac{1056598\left(\sum _{\mathrm{\_n}=9}^{\mathrm{\infty }}\frac{\left(\mathrm{\_n}-8\right)2^{\mathrm{\_n}}\left(\sum _{\mathrm{\_n1}=9}^{\mathrm{\_n}-1}\frac{{\left(\frac{1}{8}\right)}^{\mathrm{\_n1}}\mathrm{\Gamma }\left(\mathrm{\_n1}+1\right)}{\left(\mathrm{\_n1}-7\right)\left(\mathrm{\_n1}-8\right)}\right){\left(x+3\right)}^{\mathrm{\_n}}}{\mathrm{\_n}!}\right)}{12170655}-\frac{2\left(\sum _{\mathrm{\_n}=9}^{\mathrm{\infty }}\frac{\left(\mathrm{\_n}-8\right)2^{\mathrm{\_n}}\left(\sum _{\mathrm{\_n1}=9}^{\mathrm{\_n}-1}\frac{{\left(\frac{1}{8}\right)}^{\mathrm{\_n1}}\mathrm{\Gamma }\left(\mathrm{\_n1}+1\right)\left(\sum _{\mathrm{\_n2}=9}^{\mathrm{\_n1}-1}\frac{\left(\mathrm{\_n2}-7\right){\left(\mathrm{-12}\right)}^{\mathrm{\_n2}}\left(\mathrm{I}\sqrt{11}-2\mathrm{\_n2}-7\right)\left(\mathrm{I}\sqrt{11}+2\mathrm{\_n2}+7\right)}{\left(\mathrm{\_n2}+1\right)\mathrm{\Gamma }\left(\mathrm{\_n2}+1\right)}\right)}{\left(\mathrm{\_n1}-7\right)\left(\mathrm{\_n1}-8\right)}\right){\left(x+3\right)}^{\mathrm{\_n}}}{\mathrm{\_n}!}\right)}{19683\left(\mathrm{I}\sqrt{11}-25\right)\left(\mathrm{I}\sqrt{11}+25\right)}+\frac{869574607}{273468358656}+\frac{243522833x}{273468358656}+\frac{3853999{\left(x+3\right)}^2}{5064228864}+\frac{366367{\left(x+3\right)}^3}{859963392}+\frac{1269989{\left(x+3\right)}^4}{7596343296}+\frac{11983373{\left(x+3\right)}^5}{227890298880}+\frac{1166693{\left(x+3\right)}^6}{113945149440}+\frac{572503{\left(x+3\right)}^7}{265872015360}-\frac{93781{\left(x+3\right)}^8}{398808023040}\right){\mathrm{\_C}}_1\right\}$

$\mathrm{ode2}≔\mathrm{diff}\left(y\left(x\right)\,x\right)-y\left(x\right)=\mathrm{Sum}\left(\frac{\left(\mathrm{Sum}\left(\mathrm{\Gamma }\left(2\mathrm{n1}+1\right)\,\mathrm{n1}=0..n-1\right)\right)x^n}{n!}\,n=0..\mathrm{\infty }\right)$

$\mathrm{ode2}≔\frac{\ⅆ}{\ⅆx}y\left(x\right)-y\left(x\right)=\sum _{n=0}^{\mathrm{\infty }}\frac{\left(\sum _{\mathrm{n1}=0}^{n-1}\mathrm{\Gamma }\left(2\mathrm{n1}+1\right)\right)x^n}{n!}$

$\mathrm{dAlembertian\_series\_sol}\left(\mathrm{ode2}\,y\left(x\right)\,'\mathrm{outputHGT}'='\mathrm{active}'\,'\mathrm{indices}'=\left[n\,k\right]\right)$

$\left(\sum _{n=1}^{\mathrm{\infty }}\frac{\left(n-1+\left(\sum _{\mathrm{n1}=1}^{n-1}\sum _{\mathrm{n2}=1}^{\mathrm{n1}-1}\mathrm{\Gamma }\left(2\mathrm{n2}+1\right)\right)\right)x^n}{n!}\right)+{\mathrm{\_C}}_0\left(\sum _{n=0}^{\mathrm{\infty }}\frac{x^n}{n!}\right)$

The Slode[dAlembertian_series_sol] command was updated in Maple 2017.

The ode parameter was updated in Maple 2017.