Special formal series solutions of linear operator equations

The transformation which assigns to a linear operator L the recurrence satis ed by coe cient se quences of the polynomial series in its kernel is shown to be an isomorphism of the corresponding operator algebras We use this fact to help factoring q di erence and recurrence operators and to nd nice power series solutions of linear di erential equations In particular we characterize generalized hypergeometric series that solve a linear di erential equation with polynomial coe cients at an ordinary point of the equation and show that these solutions remain hypergeometric at any other ordinary point Therefore to nd all generalized hypergeometric series solutions it su ces to look at a nite number of points all the singular points and a single arbitrarily chosen ordinary point We also show that at a point x a we can have power series solutions with polynomial coe cient sequence only if the equation is singular at a non polynomial rational coe cient sequence only if the equation is singular at a Introduction and notation The method of solving linear di erential equations by means of power series has been known for centuries Here we look at formal series that are based on other polynomial sequences besides the powers and show how they can be used to reduce questions about operators of di erent types e g di erential di erence q di erence to questions about operators of a single type namely recurrence operators We consider a transformation RB which assigns to a linear operator L acting on the polynomial algebra K x its induced recurrence operator RBL The transformation is de ned in Section We show that RB is an isomorphism of the corresponding operator algebras This result is applied in Sections and to the cases of q di erence recurrence and di erential operators In particular we show how transformation RB can help factor these operators This is important because although general factorization algorithms are known they are still highly impractical Subsections and are devoted to the search for nice power series solutions in the di erential case We are interested in series with coe cients which are polynomial rational or hypergeometric in their subscript respectively Supported in part by grant IN RU from the RFBR and INTASS Supported in part by grant J of the Slovenian Ministry of Science and Technology zSupported in part by grant IN RU from the RFBR and INTASS Call a sequence cn n hypergeometric if there is a rational function R x such that cn R n cn for all large enough n If cn is hypergeometric and eventually nonzero then R x is uniquely determined and we call it the consecutive term ratio of cn Obviously a rational sequence is hypergeometric and the product of hypergeometric sequences is hypergeometric Two hypergeometric sequences an and bn are similar if there is a rational function r x such that an r n bn for all large enough n A linear combination of pairwise similar hypergeometric terms is obviously hypergeometric Also if an is hypergeometric and k a xed integer then an k is similar to an A formal power series y P n cnx n is called a generalized hypergeometric series if the sequence of coe cients cn n is hypergeometric Lemma Let y P n cnx n be a hypergeometric series and p x a polynomial Then p x y is a hyper geometric series with similar coe cients Proof Let p x Pd k ukx k and p x y P n bnx n Then