Function: rec2ortho - from three term recurrence to orthogonal polynomials,
Copyright 1996, 1997, 1998, 2005 by Tom H. Koornwinder and Rene' F. Swarttouw.
email thk@science.uva.nl

Calling Sequence:
 rec2ortho(an,bn,cn,A0,B0,C1)

Parameters:
 an,bn,cn - coeficients in three term recurrence for general n
 A0,B0,C1 - optional coefficients in three term recurrence for n=0 or 1

Version: 1b3 (for Maple 6, 7, 8 or 9), 19 March 2005.

Description:

* Let the sequence of polynomials p(n,x) of degree n in x be generated by
  the recurrence
  
    x*p(n,x) = an*p(n+1,x) + bn*p(n,x) + cn*p(n-1,x)
    
  The function tries to return p(n,x) in the form
  
    coef(n) * named_polynomial(n, u*x+v, parameters)
    
  with named_polynomial being a Hermite, Charlier, Laguerre, Meixner,
  Krawtchouk, Meixner-Pollaczek or Jacobi polynomial, provided u, v and
  the parameters (possibly under further assumptions) admit a positive
  orthogonality measure for the polynomial p(n,x). if such a form is not
  possible then the function returns FAIL. The function can thus recognize
  all systems of orthogonal polynomials in the Askey scheme up to the
  two-parameter level
  
* For the named orthogonal polynomials standard notation (linearized)
  and standard normalization will be followed (see [1, Appendix] or [2]):
  
    Hermite:           H(n,x)
    Charlier:          C(n,x,a)             a>0
    Laguerre:          L(n,x,alpha)         alpha>-1
    Meixner-Pollaczek: P(n,x,lambda,phi)    lambda>0, 0<phi<Pi
    Meixner:           M(n,x,beta,c)        beta>0, 0<c<1 or c>1
    Krawtchouk:        K(n,x,p,N)           0<p<1, N=1,2,..., n=0,1,...,N
    Jacobi:            P(n,x,alpha,beta)    alpha>-1, beta>-1

  In the Meixner-Pollaczek case the normalizations of [1] and [2] differ.
  We follow [2]:

    P(n,x,lambda,phi)=(2*sin(phi)^n/n!)*x^n + ...

*  Always p(0,x):=1. For n=0 the above recurrence is
     
     x = A0*p(1,x) + B0
     
   If A0 is not given then A0:=subs(n=0,an).
   If B0 is not given then B0:=subs(n=0,bn).
   If C1 is given then, for n=1, cn is replaced by C1.
   Only changing A0,B0 or C1 will highly influence the output.
   
* n and x are global variables.

* an,bn,cn may depend on n.
  A0,B0,C1 are not allowed to depend on n.
  an,bn,cn,A0,B0,C1 may depend on other parameters.
  Properties of these parameters may be assumed with the assume facility.
  rec2ortho may give additional properties to these parameters in order
  to guarantee the existence of a positive orthogonality measure.
  If a is a parameter occurring in the output and if the output mentions
  that there are assumptions on a, then these assumptions can be obtained
  with the command:    about(a);

* The tilde notation for variables with assumed properties is avoided, since
  rec2ortho puts:   interface(showassumed=2):

* In the output it is silently assumed that x is real and that n is nonnegative
  integer. Moreover, in the Krawtchouk case K(n,x,p,N), N is explicitly
  assumed to be positive integer, and it is silently assumed that
  n=0,1,...,N.

* A0 should be nonzero and an should be nonzero for positive integer n.
  Then the ouput coefficient coeff(n) will be nonzero for nonnegative integer n.
  This will not be checked exhaustively by rec2ortho.
  The user should already check this in advance.
  Possibly make assumptions on parameters such that an and A0 do not vanish.
  
* Strictly speaking, the output coefficient coef(n) should also be real.
  This will not be checked by rec2ortho.
  
* Often, because of symmetry transformations for the named orthogonal
  polynomials, two different forms of the output are possible. It cannot
  be predicted which of the two forms rec2ortho will give.
  
* In some cases it will depend on some quantity being positive, zero
  or negative whether we may be in the Meixner-Pollaczek case, Laguerre case
  or Meixner and Krawtchouk case, respectively. In the third case the sign of
  another quantity will decide between the Meixner and Krawtchouk case.
  rec2ortho will tell that the user should make such a choice (by the
  assume facility) and then rerun rec2ortho.

References:

[1] R. Askey and J.A. Wilson, Some basic hypergeometric orthogonal
    polynomials that geenralize Jacobi polynomials,
    Memoirs Amer. Math. Soc., No.319 (1985).
    
[2] R. Koekoek and R.F. Swarttouw, The Askey scheme of hypergeometric
    orthogonal polynomials and its q-analogue,
    Delft University of Technology, Faculty of
    Information Technology and Systems, Report 98-17, 1998;
    online: http://aw.twi.tudelft.nl/~koekoek/askey/
    
    
Examples:

> read rec2ortho:
> rec2ortho(1/2,0,n);# Hermite

                               Hermite
                               H(n, x)

> rec2ortho(2*(n+1)*(n+alpha+beta+1)/(2*n+alpha+beta+1)/(2*n+alpha+beta+
> 2),(beta^2-alpha^2)/(2*n+alpha+beta)/(2*n+alpha+beta+2),2*(n+alpha)*(n
> +beta)/(2*n+alpha+beta)/(2*n+alpha+beta+1));# Jacobi

                              Jacobi
                        P(n, x, alpha, beta)
                with assumptions on alpha and beta

> about(alpha); about(beta);
Originally alpha, renamed alpha~:
  is assumed to be: RealRange(Open(-1),infinity)

Originally beta, renamed beta~:
  is assumed to be: RealRange(Open(-1),infinity)

> rec2ortho(-p*(N-n),p*(N-n)+n*(1-p),-n*(1-p));# Krawtchouk

    possibly Meixner or Krawtchouk, assume more about the sign of
                                  -N
                                 FAIL

> assume(N>0):
> rec2ortho(-p*(N-n),p*(N-n)+n*(1-p),-n*(1-p));# Krawtchouk again

                             Krawtchouk
                           K(n, x, p, N)
                   with assumptions on p and N
                   
> about(N); about(p);
Originally N, renamed N~:
  is assumed to be: AndProp(RealRange(1,infinity),integer)

Originally p, renamed p~:
  is assumed to be: RealRange(Open(0),Open(1))

> rec2ortho(1,-(n+lambda)*c,n*(n+2*lambda-1)/4);

possibly Meixner or Krawtchouk or Meixner-Pollaczek,
    assume more about the sign of
                                 2                                
                                c  - 1
                                 FAIL

# Meixner-Pollaczek (-1<c<1), Laguerre(c=1), Meixner or Krawtchouk (c>1)


Files:
  rec2ortho (source code)
  rec2orthotest.mws (Maple worksheet for Maple 6, 7, 8
                                                with many further examples)
  rec2orthotest.mw (idem for Maple 9)