From: fwchapma@daisy.uwaterloo.ca (Frederick W. Chapman) Newsgroups: sci.math.research Subject: Linear Diff Eqns w/Polynomial Coeffs Date: Wed, 21 Aug 1996 02:11:50 GMT (0) NOTATION ------------ Let D := d/dx, and let R denote the real numbers. Let R[D] denote polynomials in the operator D with coefficients in R, and R[x,D] denote polynomials in the operator D with coefficients in R[x]. (1) WHAT I KNOW SO FAR ---------------------- Let K := { u(x) | there exists p(D) in R[D] so that p(D) u(x) = 0 }. That is, K consists of ALL functions which are annihilated by SOME linear differential operator with CONSTANT coefficients. Thus, every element of K is a finite linear combination of polynomial-exponential functions. I know that: (1.1) K is an infinite-dimensional vector space. (1.2) K is closed under pointwise multiplication. (1.3) The Laplace transforms maps K bijectively onto the set of PROPER rational functions with real coefficients. Despite all these nice properties, the function algebra K is not large enough for my purposes since its elements are too well-behaved -- all elements of K are analytic on the whole real line. K contains no rational or algebraic functions other than polynomials, for example. (2) WHAT I'D LIKE TO FIND OUT ----------------------------- Let L := { u(x) | there exists p(x,D) in R[x,D] with p(x,D) u(x) = 0 }. That is, L consists of ALL functions which are annihilated by SOME linear differential operator with POLYNOMIAL coefficients. L is clearly a superset of K, and admits rational functions and algebraic functions which are not analytic on the whole real line. In addition, L contains almost all the special functions of mathematical physics. This is what I would like to know: (2.1) Is L a vector space? (2.2) Is L closed under pointwise multiplication? (2.3) Do the elements of L have well-defined Laplace transforms, and do those Laplace tranforms have a simple characterization? These questions are harder to resolve for L than for K because the differential operators which define L are not autonomous and do not commute. Any answers and/or references to the literature would be greatly appreciated! Thanks very much, Fred Chapman ---------------------------------------------------------------------------- Frederick W. Chapman, University of Waterloo (fwchapman@daisy.uwaterloo.ca) - Department of Applied Mathematics MC 4008 (519) 888-4567, x5917 - Symbolic Computation Group/Maple Lab DC 2302E (519) 888-4567, x4474 Applied Mathematics, University of Waterloo, Waterloo, Ont., N2L 3G1, CANADA -- Frederick W. Chapman, University of Waterloo (fwchapman@daisy.uwaterloo.ca) - Department of Applied Mathematics MC 4008 (519) 888-4567, x5917 - Symbolic Computation Group/Maple Lab DC 2302E (519) 888-4567, x4474 Applied Mathematics, University of Waterloo, Waterloo, Ont., N2L 3G1, CANADA ============================================================================== From: wgd@berne.ai.mit.edu (Bill Dubuque) Newsgroups: sci.math.research Subject: Re: Linear Diff Eqns w/Polynomial Coeffs Date: 27 Aug 1996 02:36:15 -0400 See papers of L. Rubel, M. Singer, R. Stanley, D. Zeilberger for a start. Deeper results are found by studying D-modules and holonomic functions. -Bill ============================================================================== From: rusin@vesuvius.math.niu.edu (Dave Rusin) Newsgroups: sci.math.research Subject: Re: Linear Diff Eqns w/Polynomial Coeffs Date: 28 Aug 1996 06:27:48 GMT In article , Frederick W. Chapman wrote: > >Let D := d/dx, and let R denote the real numbers. Let R[D] denote >polynomials in the operator D with coefficients in R, and R[x,D] denote >polynomials in the operator D with coefficients in R[x]. > >Let L := { u(x) | there exists p(x,D) in R[x,D] with p(x,D) u(x) = 0 }. > >That is, L consists of ALL functions which are annihilated by SOME linear >differential operator with POLYNOMIAL coefficients. L is clearly a >superset of K, and admits rational functions and algebraic functions >which are not analytic on the whole real line. In addition, L contains >almost all the special functions of mathematical physics. This is what I >would like to know: You need "nonzero" in front of p(x,D), of course. I assume you have already noted that p can be simplified to a sum of (polynomials in x) times (powers of D), that is, no mixed products (e.g. xDxD ) are needed. I suppose one should object that the domains are too vague here. One could imagine elements of L defined respectively on ( - oo, 0) and ( 0 , oo ); what then is their sum? But I'm willing to treat these a little formally. Dividing by the coefficient of the highest power of D, we see that L is the set of functions u such that the R(x)-linear span S(u) of {u, u', u'', ...} is finite-dimensional. Then the answer to >(2.1) Is L a vector space? (over R , I assume) is "yes" -- scalar multiplcation is trivial, and if S(u) = span{u, u', ..., u^(n)} and S(v) = span{v, v', ..., v^(m)}, then S(u+v) is contained in the span of all (n+1) + (m+1) derivatives u^(i)+v^(j), and hence is finite-dimensional too: u+v lies in L. Likewise, it's easy to show that all derivatives of u.v are linear combinations of the functions u^(i).v^(j) and hence are sums of elements in the finite-dimensional space S(u) \tensor_{R(x)} S(v), so that S(u.v) is also finite-dimensional over R(x), and the answer to >(2.2) Is L closed under pointwise multiplication? is also "yes". >(2.3) Do the elements of L have well-defined Laplace transforms, and do >those Laplace tranforms have a simple characterization? Dunno. >Any answers and/or references to the literature would be greatly >appreciated! Your objects of study seem very natural; surely the arguments I just gave are in the literature somewhere, but I have no idea where. dave ============================================================================== From: Alan Horwitz Newsgroups: sci.math.research Subject: Re: Linear Diff Eqns w/Polynomial Coeffs Date: Fri, 30 Aug 1996 20:48:57 -0700 Dave Rusin wrote: > > In article , > Frederick W. Chapman wrote: > > > >Let D := d/dx, and let R denote the real numbers. Let R[D] denote > >polynomials in the operator D with coefficients in R, and R[x,D] denote > >polynomials in the operator D with coefficients in R[x]. > > > >Let L := { u(x) | there exists p(x,D) in R[x,D] with p(x,D) u(x) = 0 }. > > > >That is, L consists of ALL functions which are annihilated by SOME linear > >differential operator with POLYNOMIAL coefficients. L is clearly a > >superset of K, and admits rational functions and algebraic functions > >which are not analytic on the whole real line. In addition, L contains > >almost all the special functions of mathematical physics. This is what I > >would like to know: > > You need "nonzero" in front of p(x,D), of course. I assume you have > already noted that p can be simplified to a sum of (polynomials in x) > times (powers of D), that is, no mixed products (e.g. xDxD ) are needed. > > > > > >Any answers and/or references to the literature would be greatly > >appreciated! > > Your objects of study seem very natural; surely the arguments I just > gave are in the literature somewhere, but I have no idea where. > > dave There is an interesting paper by R.P.Stanley called "Differentiably Finite Power Series" in Europ.J.Combinatorics(1980)1,175-188. He discusses in length many of the properties of L. One idea I had, but never did anything with, is the following: Let L1=L, L2=set of all solutions of linear differential equations with coefficients in L, etc. Can one characterize lim(Ln) ? In particular, does it equal the differentially algebraic functions-i.e. solutions of P(x,y,y',...,y^(n))=0, where P is a polynomial in all of its variables?