Computing Hypergeometric Solutions of Linear Recurrence Equations

We describe a complete algorithm to compute the hypergeometric solutions of linear recurrence relations with rational function coefficients. We use the notion of finite singularities and avoid computations in splitting fields. An implementation is available in Maple 9.

[1]  C. R. Adams,et al.  On the irregular cases of the linear ordinary difference equation , 1928 .

[2]  Sergei A. Abramov.  Rational solutions of linear difference and q-difference equations with polynomial coefficients , 1995, ISSAC '95.

[3]  Marius van der Put,et al.  Differential equations in characteristic p , 1995 .

[4]  Sergei A. Abramov,et al.  q-Hypergeometric solutions of q-difference equations , 1998, Discret. Math..

[5]  Marius van der Put,et al.  Galois Theory of Difference Equations , 1997 .

[6]  Mark van Hoeij,et al.  A modular algorithm for computing the exponential solutions of a linear differential operator , 2004, J. Symb. Comput..

[7]  Norlund N.-E..  Lecons sur les equations lineaires aux differences finies , 1929 .

[8]  A. Barkatou,et al.  Sur les séries formelles solutions d'équations aux différences polynomiales , 1994 .

[9]  Moulay A. Barkatou,et al.  Rational solutions of first order linear difference systems , 1998, ISSAC '98.

[10]  Mark van Hoeij,et al.  Finite singularities and hypergeometric solutions of linear recurrence equations , 1999 .

[11]  Alin Bostan,et al.  Fast algorithms for polynomial solutions of linear differential equations , 2005, ISSAC.

[12]  Moulay A. Barkatou.  Rational Solutions of Linear Differential Systems , 2006 .

[13]  George D. Birkhoff,et al.  Formal theory of irregular linear difference equations , 1930 .

[14]  Manuel Bronstein,et al.  On polynomial solutions of linear operator equations , 1995, ISSAC '95.

[15]  Translation Program. , 1960, Science.

[16]  Emanuel Beke,et al.  Die Irreducibilität der homogenen linearen Differentialgleichungen , 1894 .

[17]  Moulay A. Barkatou,et al.  Apparent singularities of linear difference equations with polynomial coefficients , 2005, Applicable Algebra in Engineering, Communication and Computing.

[18]  Mark Giesbrecht,et al.  Factoring and decomposing ore polynomials over Fq(t) , 2003, ISSAC '03.

[19]  Mark van Hoeij,et al.  Rational solutions of linear difference equations , 1998, ISSAC '98.

[20]  Marko Petkovsek,et al.  Hypergeometric Solutions of Linear Recurrences with Polynomial Coefficents , 1992, J. Symb. Comput..