Expansion of polynomials

Parameters

The expansion of (exponential) generating functions is performed by PolynomialFieldSequence with constructors that have the following parameters:

• offset: first index
• polys: a list of list of integers in square brackets, specifying the coefficients of a list of polynomials.
• postfix: a string starting with a delimiter, followed by a list of elements that are separated by this delimiter
• dist: an optional increment to the current index
• gfType: 0 for an o.g.f., 1 for an e.g.f., +4 if the denominator should be returned instead of the numerator (default 0 = o.g.f)
• modulus, periodicity: which element to take when zeros are to be suppressed. For example, even coefficients are selected by (0,2) and odd coefficients by (1,2).
• seqs: a variable number of sequences defining subordinate generating functions

Generator Callcodes

The following callcodes of the generator use the specified parameter combinations:

poly    (offset, polys, postfix)
polyx   (offset, polys, postfix, dist, gftype)
polyz   (offset, polys, postfix, dist, gftype, modulus, factor)
polya   (offset, polys, postfix, seqs)
polyxa  (offset, polys, postfix, dist, gftype, seqs)
polyza  (offset, polys, postfix, dist, gftype, modulus, factor, seqs)

Polynomials

The polynomials in polys are referenced by p0, p1, p2, ... where p0 gives the initial term of the iteration. A single polynomial is specified by the list of coefficients with increasing exponents, for example, 1-x = 1*x^0 -1*x^1 -> [1,-1]. Most often the initial p0 = "[1]" = 1. The list of polynomials may optionally (for testing) be followed by a list of sequence numbers for subordinate generating functions, but normally such sequences should be specified as a separate trailign list of sequence instances.

Examples:

• A308847 Expansion of e.g.f. exp(-2*x) / BesselI(0,2*x).

 A-number  callcode offset  p0=1,p1=2*x,p2=-2*x  exp(p2)/besselI(0,p1)  dist  gftype=exp
 A308847   polyx    0      "[1],[0,2],[0,-2]"   "p2,exp,0,p1,besselI,/" 0     1

• A345969 Expansion of the e.g.f. 1 / sqrt(3 - 2 / ((1 - x)*exp(x)))
  • Maple: a := series(1/sqrt(3-2/((1-x)*exp(x))), x=0, 25): seq(n!*coeff(a, x, n), n=0..24);
  • jOEIS: new PolynomialFieldSequence(0, "[[1],[1,-1]]" ",1,3,2,p1,x,exp,*,/,-,sqrt,/", 0, 1)

Postfix notation

The operations of the postfix string are performed in their sequential order. They operate on a stack, and the final result is the lowest stack element.

Special operators

• A push the generating function to be defined
• S, T, U, V push the 1st, 2nd, 3rd, 4th generating function from the list of additional sequences
• x push the free variable
• 0, 1, 2, ... push an integer
• n push the current index
• pi push the predefined polynomial with index i (cf. polys above)
• <i shift the top stack element by i, i.e. multiply by xⁱ
• ^i take the top stack element to the i-th power, where i can be an integer or a fraction (^1/2 is sqrt)
• dif differentiate the top stack element
• int formal integration of the top stack element
• sub substitute, call the g.f. A with the top stack element

Arithmetic operators, functions with 2 arguments

Replace the 2 top stack elements a, b by a op b, where op is one of:

• + - * /
• agm
• besselI

Functions with 1 argument

Replace the top stack element by the function's result:

• abs acosh asin asinh atan atanh cos cosh cot coth csc csch ellipticD ellipticE ellipticK eta exp lambertW log neg sec sech sin sinh sqrt tan tanh
• catalan(x) = (1 - sqrt(1 - 4*x)) / (2*x)

Subordinate generating functions

Any sequence implemented in jOEIS can be used as a subordinate generating function. Usual examples are the Eta Products and theta_3 = A000122, theta_4 = A002448.

Caveats

• Usually the g.f.s have offset 0 and start with a term 1, so p0 = "1" in most cases. The first polynomial p0 must always include the coefficient of x⁰, even if the offset is > 0, for example: p0 = "0,1".
• Usually the parameter dist is not needed. Sometimes, it must be set to a value > 0 to start the iteration properly.
• sqrt expects an argument of the form 1-x. sqrt(x) does not work, it can sometimes be avoided by quadrating all occurrences of x.

Commandline activation

A script OEIS-mat/scripts/poly facilitates the testing of the polynomial expansions. It is activated with the following parameters and options:

poly polys postfix [-i dist] [-t gfTtype] [-n number of terms] [-o offset] [-d debug mode]