basis.mpl

Hypergeometric series solutions of linear operator equations

• FindBase(L,x,n,{pars},[bas,t_par])

  Search for suitable basis up to a given degree.

  Parameters:

  L: the linear operator defined as a function. For example,

  L:=proc(f,x) subs(x=x+1,f)-f end:

  x: the variable which the operator acts on.

  n: the maximal degree of the basis polynomials to compute.

  pars: parameters that we want the basis being independent of. Usually take the empty set.

  Optimal parameters:

  bas: the lattice x(s). For example, bas=s*(s+u)

  t_par: the parameter in x(s) that need to be determined. For example, t_par={u}.

  The output is a list of solutions. The first element indicate x(s). The list follows indicate the root x_k.

  An example:

  Wilson polynomials:

  The operator (difference equation) is given by

  L:=proc(f,s) local bs,ds;
     bs:=(a-I*s)*(b-I*s)*(c-I*s)*(d-I*s)/(2*I*s*(2*I*s-1));
     ds:=(a+I*s)*(b+I*s)*(c+I*s)*(d+I*s)/(2*I*s*(2*I*s+1));
     simplify( n*(n+a+b+c+d-1)*f - (bs*subs(s=s+I,f) -(bs+ds)*f + ds*subs(s=s-I,f)) );
  end proc:

  Now use the commond

  FindBase(L,s,5,{},bas=s*(s+u),t_par={u});

  We get

  {[1, [s^2, [-a^2, -(a+1)^2, -(a+2)^2, -(a+3)^2, -(a+4)^2]], [s^2, [-b^2, -(b+1)^2, -(b+2)^2, -(b+3)^2, -(b+4)^2]], [s^2, [-c^2, -(c+1)^2, -(c+2)^2, -(c+3)^2, -(c+4)^2]], [s^2, [-d^2, -(d+1)^2, -(d+2)^2, -(3+d)^2, -(d+4)^2]]]}

  which means that one suitable basis is

  x(s)=s^2,    x_1=-a^2, x_2=-(a+1)^2, x_3=-(a+2)^2, ...

  Another suitable basis is

  x(s)=s^2,    x_1=-b^2, x_2=-(b+1)^2, x_3=-(b+2)^2, ...

  And so on.

  Discrete q-Hermite II

  The operator (three term recurrece) is given by

  L:=proc(f,qn) ( x*f - ( I/qn* subs(qn=qn*q,f) - I/qn*(1-qn)*subs(qn=qn/q,f) )); end proc:

  The command FindBase(L,qn,5,{}); returns

  {[1, [qn, [0, 0, 0, 0, 0]], [qn, [1, q, q^2, q^3, q^4]]]}

• Verify(L,x,bk,k,h)

  Verfiy whether a sequence of polynomials is a suitable basis.

  Parameters:

  L: the linear operator

  x: the variable

  bk: the basis polynomial of degree k

  k: the degree varible k

  h: the depth of the recurrence L(b_k)=A b_k + B b_{k-h}

  If bk is a suitable basis, then return A_k/B_{k+h}

  An example: Hahn polynomials

  L:=proc(f,x)

  local bx,dx;

     bx:=(x+alpha+1)*(x-N);

     dx:=x*(x-beta-N-1);

     simplify( n*(n+alpha+beta+1)*f - (bx*subs(x=x+1,f) -(bx+dx)*f + dx*subs(x=x-1,f)) );

  end proc:

  We want to verify x(x-1)...(x-k+1) is a suitible basis.

  Verify(L,x,(-1)^k*pochhammer(-x,k),k,1);

  [(k-n)*(k+n+alpha+beta+1)/(k+1)/(alpha+k+1)/(N-k)]

• qVerify(L,x,bk,k,h)

  The q-analogue of Verify. We need Koepf's package qsum9.mpl to simplify the expression.

• Findr(alpha,beta,gamma,n)

  List all candidates for r(n) (Section 5)

  Parameters: alpha,beta,gamma are the coefficients in the three term recurrence relations. n is the variable.

• qFindr(alpha,beta,gamma,t)

  The q-ananlogue of Findr, where t=q^{-n}.