Sergei A. Abramov
ABSTRACT
D'Alembertian solutions of differential (resp. difference) equations are those expressible as nested indefinite integrals (resp. sums) of hyperexponential functions. They are a subclass of Liouvillian solutions, and can be constructed by recursively finding hyperexponential solutions and reducing the order. Knowing d'Alembertian solutions of Ly = 0, one can write down the corresponding solutions of Ly = f and of L*y = 0.
- Abr89.S.A. Abramov (1989): Rational solutions of linear difference and differential equations with polynomiM coefficients, USSR Comput. Maths. Math. Phys. 29, 7-12, transl, from Zl. Vychislit. matem, mat. fiz. 29, 1611-1620. Google Scholar
Digital Library
- Abr&Kva91.S.A. Abramov, K.Yu. Kvashenko (1991): Fast Mgorithm to search for the rational solutions of linear differential equations, Proc. ISSA C'91, 267-270. Google ScholarDigital Library
- Abr93.S.A. Abramov (1993): On d'Alembert substitution, Proc. ISSA C'93, 20-26. Google ScholarDigital Library
- Abr94a.S.A. Abramov (1993): D'Alembert substitution and adjoint equation (computer algebra aspect), Cornput. Maths. Math. Phys. 34, no. 7. In print. Google ScholarDigital Library
- Abr94b.S.A. Abramov (1994): D'Alembertian solutions of nonhomogeneous linear ordinary differential and difference equations. To be published.Google Scholar
- Alm&Zei90.G.Almkvist, D.Zeilberger (1990): The method of differentiating under the integral sign, .1. Symb. Comp. 10, 571-591. Google ScholarDigital Library
- Bro92a.M. Bronstein (1992): On solutions of linear ordinary differential equations in their coefficient field, J. Symb. Comp. 13, 413-439. Google ScholarDigital Library
- Bro92b.M. Bronstein (1992): Linear differential equations: breaking through the order 2 barrier, Proc. ISSA C'92, 42-48. Google ScholarDigital Library
- Bro&Pet94.M. Bronstein, M. Petkov~ek (1994): On Ore rings, linear operators and factorisation, Programmirovanie 1, no. 1. To appear.Google Scholar
- Gos78.R.W. Gosper, Jr. (1978): Decision procedure for indefinite hypergeometric summation, Proc. Natl. A cad. Sci. USA 75, 40-42.Google ScholarCross Ref
- Koo91.R.J. Kooman (1991): Convergence properties of recurrence sequence, Centrum voor Wiskunde en Informatica, CWI Tract 83, Amsterdam.Google Scholar
- Ore33.O. Ore (1933): Theory of non-commutative polynomials, Ann. Maths. 34, 480-508.Google ScholarCross Ref
- Pet92.M. Petkov.~ek (1992): Hypergeometric solutions of linear recurrences with polynomial coefficients, J. Symb. Comp. 14, 243-264. Google ScholarDigital Library
- Sin81.M.F. Singer (1981): Liouvillian solutions of n-th order homogeneous linear differential equations, Amer. J. Math. 103, 661-681.Google ScholarCross Ref
- Sin91.M.F. Singer (1991): Liouvillian solutions of linear differential equations with Liouvillian coefficients, J. Symb. Comp. 11, 251-273. Google ScholarDigital Library
Index Terms
- D'Alembertian solutions of linear differential and difference equations
Recommendations
Polynomial solutions of differential-difference equations
We investigate the zeros of polynomial solutions to the differential-difference equation P"n"+"1=A"nP"n^'+B"nP"n,n=0,1,... where A"n and B"n are polynomials of degree at most 2 and 1 respectively. We address the question of when the zeros are real and ...
Linear partial q-difference equations on q-linear lattices and their bivariate q-orthogonal polynomial solutions
Orthogonal polynomial solutions of an admissible potentially self-adjoint linear second-order partial q-difference equation of the hypergeometric type in two variables on q-linear lattices are analyzed. A q-Pearson's system for the orthogonality weight ...
On polynomial solutions of linear partial differential and (q-)difference equations
CASC'12: Proceedings of the 14th international conference on Computer Algebra in Scientific ComputingWe prove that the question of whether a given linear partial differential or difference equation with polynomial coefficients has non-zero polynomial solutions is algorithmically undecidable. However, for equations with constant coefficients this ...
Comments