FUNCTION: ToQPh - express any quasi-symmetric function in the QPh-basis (dual basis of noncommutative power sums symmetric function of the second kind)

CALLING SEQUENCE:

ToQPh(qsf)

ToQPh(qsf, b)

NCSF[ToQPh](qsf)

NCSF[ToQPh](qsf, b)

PARAMETERS:

qsf = any quasi-symmetric function

b = any name of a known basis

SYNOPSIS:

• The ToQPh function computes the expansion of qsf in the QPh-basis.

• The input is any expression in terms of the basic quasi-symmetric functions.

• The quasi-symmetric function qsf is expanded and the result is not collected.

• One may specify by a second argument, say b, that qsf is solely expressed in terms of the known basis b.

• The call ToQPh(qsf, 'QPh') does not affect the argument qsf.

• One may add 'noexpand' just after the argument qsf to choose not to expand the quasi-symmetric function qsf before treating it.

• One may collect the result by adding a third argument: this is done by ToQPh(qsf, b, 'collect'). For instance, ToQPh(qsf, 'QPh', 'collect') may be used to collect the argument qsf.

• The noncommutative multiplication is denoted by the &* operator.

• Whenever there is a conflict between the function name ToQPh and another name used in the same session, use the long form NCSF['ToQPh'].

EXAMPLES:

> with(NCSF):
> ToQPh((1+q)^2*QPs[2,1]):                       # expands the input
> ToQPh((1+q)^2*QPs[2,1],noexpand):              # does not expand (1+q)^2
> ToQPh((1+q)^2*QPs[2,1],collect):               # collects the result
> ToQPh((1+q)^2*QPh[2,1]*QPh[2],'QPh'):          # specifies a basis
> ToQPh((1+q)^2*QPh[2,1],noexpand,'QPh',collect):# the most efficient
> ToQPh((1+q)^2*QPs[2,1]+M[3,1]+t^3*M[2,2],noexpand,collect);
 
                       2                   2              3
          - 1/2 (1 + q)  QPh[3] + 3 (1 + q)  QPh[2, 1] + t  QPh[2, 2]
 
                             3
             + (- 1/2 - 1/2 t ) QPh[4] + QPh[3, 1]