{\displaystyle z^{2}u''+p(z)zu'+q(z)u=0,}

 

{\displaystyle u'\equiv {{du} \over {dz))}   و   {\displaystyle u''\equiv {{d^{2}u} \over {dz^{2))))

 

{\displaystyle u(z)=\sum _{k=0}^{\infty }A_{k}z^{k+r},\qquad (A_{0}\neq 0).}

 

{\displaystyle u'(z)=\sum _{k=0}^{\infty }(k+r)A_{k}z^{k+r-1},}

{\displaystyle u''(z)=\sum _{k=0}^{\infty }(k+r-1)(k+r)A_{k}z^{k+r-2},}

 

{\displaystyle z^{2}\sum _{k=0}^{\infty }(k+r-1)(k+r)A_{k}z^{k+r-2}+zp(z )\sum _{k=0}^{\infty }(k+r)A_{k}z^{k+r-1}+q(z)\sum _{k=0}^{\infty } A_{k}z^{k+r}=}

{\displaystyle =\sum _{k=0}^{\infty }(k+r-1)(k+r)A_{k}z^{k+r}+p(z)\sum _{k =0}^{\infty }(k+r)A_{k}z^{k+r}+q(z)\sum _{k=0}^{\infty }A_{k}z^{k +r}=}

{\displaystyle =\sum _{k=0}^{\infty }[(k+r-1)(k+r)A_{k}z^{k+r}+p(z)(k+r )A_{k}z^{k+r}+q(z)A_{k}z^{k+r}]=}

{\displaystyle =\sum _{k=0}^{\infty }\left[(k+r-1)(k+r)+p(z)(k+r)+q(z)\right] A_{k}z^{k+r}=}

{\displaystyle =\left[r(r-1)+p(z)r+q(z)\right]A_{0}z^{r}+\sum _{k=1}^{\infty } \ چپ[(k+r-1)(k+r)+p(z)(k+r)+q(z)\right]A_{k}z^{k+r}.}

 

{\displaystyle r\left(r-1\right)+p\left(0\right)r+q\left(0\right)=I(r)}

 

{\displaystyle I(k+r)A_{k}+\sum _{j=0}^{k-1}{(j+r)p^{(kj)}(0)+q^{(kj )}(0) \over (kj)!}A_{j}.}

 

{\displaystyle I(k+r)A_{k}+\sum _{j=0}^{k-1}{(j+r)p^{(kj)}(0)+q^{(kj )}(0) \over (kj)!}A_{j}=0،}

{\displaystyle \sum _{j=0}^{k-1}{(j+r)p^{(kj)}(0)+q^{(kj)}(0) \over (kj)! }A_{j}=-I(k+r)A_{k}،}

{\displaystyle {1 \over -I(k+r)}\sum _{j=0}^{k-1}{(j+r)p^{(kj)}(0)+q^{( kj)}(0) \over (kj)!}A_{j}=A_{k}.}

 

{\displaystyle U_{r}(z)=\sum _{k=0}^{\infty }A_{k}z^{k+r))

 

{\displaystyle z^{2}U_{r}(z)''+p(z)zU_{r}(z)'+q(z)U_{r}(z)=I(r)z^{ r}.}