The class transcendental_sum_type_C involves a conjugation. More...

#include <transcendental_C.h>

Inheritance diagram for transcendental_sum_type_C:

Public Member Functions
	transcendental_sum_type_C (const GiNaC::ex &nn, const GiNaC::ex &i, const GiNaC::ex &l, const GiNaC::ex &v, const GiNaC::ex &ss, const GiNaC::ex &eps, int o, int f)

void	archive (GiNaC::archive_node &node) const override

void	read_archive (const GiNaC::archive_node &node, GiNaC::lst &sym_lst) override

unsigned	return_type (void) const override

void	print (const GiNaC::print_context &c, unsigned level=0) const override

unsigned	precedence (void) const override

GiNaC::ex	eval () const override

GiNaC::ex	subs (const GiNaC::exmap &m, unsigned options=0) const override

virtual GiNaC::ex	eval_explicit () const

virtual unsigned	get_key (void) const

virtual GiNaC::ex	hash_data (void) const

virtual GiNaC::ex	subst_data (void) const

GiNaC::ex	set_expansion (void) const

GiNaC::ex	distribute_over_subsum (void) const

GiNaC::ex	distribute_over_letter (void) const

GiNaC::ex	shift_plus_one (void) const

GiNaC::ex	shift_minus_one (void) const

Protected Member Functions
GiNaC::ex	eval_ncmul (const GiNaC::exvector &v) const override

GiNaC::ex	derivative (const GiNaC::symbol &s) const override

unsigned	calchash (void) const override

Protected Attributes
GiNaC::ex	n

GiNaC::ex	index

GiNaC::ex	letter

GiNaC::ex	lst_of_gammas

GiNaC::ex	subsum

GiNaC::ex	expansion_parameter

int	order

int	flag_expand_status

Detailed Description

The class transcendental_sum_type_C involves a conjugation.

The definition is

$\frac{\Gamma(1+d_1\varepsilon)}{\Gamma(1+b_1\varepsilon)} \frac{\Gamma(1+d_2\varepsilon)}{\Gamma(1+b_2\varepsilon)} ... \frac{\Gamma(1+d_k\varepsilon)}{\Gamma(1+b_k\varepsilon)}$

$\mbox{} (-1) \sum\limits_{i=1}^n \left( \begin{array}{c} n \\ i \\ \end{array} \right) \left( -1 \right)^i \frac{x^i}{(i+c)^m} \frac{\Gamma(i+a_1+b_1\varepsilon)}{\Gamma(i+c_1+d_1\varepsilon)} \frac{\Gamma(i+a_2+b_2\varepsilon)}{\Gamma(i+c_2+d_2\varepsilon)} ... \frac{\Gamma(i+a_k+b_k\varepsilon)}{\Gamma(i+c_k+d_k\varepsilon)} S(i+o,m_1,...,m_l,x_1,...,x_l)$

Here, , and all and are integers. In addition is non-negative or $c \ge o$ .

The upper summation limit should not be infinity.

Member Function Documentation

◆ distribute_over_letter()

ex distribute_over_letter ( void ) const

letter is allowed to contain a sum of products (e.g. an expression in expanded form). Each term can contain scalars and basic_letters.

This routine converts the transcendental_sum_type_C to a canonical form, so that afterwards letter only contains a basic_letter.

◆ distribute_over_subsum()

ex distribute_over_subsum ( void ) const

subsum is allowed to contain a sum of products (e.g. an expression in expanded form). Each term can contain scalars, basic_letters, list_of_tgammas, Zsums or Ssums.

This routine converts the transcendental_sum_type_C to a canonical form, so that afterwards subsum only contains a Ssum.

The algorithm is based on the following steps:

If an object is of type basic_letter and the difference of its index with the index member is an integer, it is combined with the letter member. Otherwise it is taken out of the sum.
If an object is of type list_of_tgamma and the difference of its index with the index member is an integer, it is combined with the lst_of_gammas member. Otherwise it is taken out of the sum.
If an object is of type Ssum and the difference of its index with the index member is an integer, it is stays inside the subsum. Otherwise it is taken out of the sum.
If an object is of type Zsum and the difference of its index with the index member is an integer, it is converted to an Ssum and stays inside the subsum. Otherwise it is taken out of the sum.

◆ eval()

ex eval ( ) const

override

Simplifications, which are always performed are:

If subsum is equal to 1, the subsum is replaced with S(i).
If subsum is not of type Ssum, the routine distribute_over_subsum is called.
If the difference of the upper summation index of the Ssum with the index member is not an integer, the Ssum is taken out of the sum. The subsum is replaced with S(i).
If letter is not of type basic_letter, the routine distribute_over_letter is called.
If the difference of the index of the basic_letter with the index member is not zero, the index in basic_letter is adjusted.
If the difference of the index of lst_of_gammas with the index member is not zero, the index in lst_of_gammas is adjusted.

If flag_expand_status == expand_status::expansion_required, the evaluation routine performs a set of consistency checks:

It adjusts the upper summation limit in subsum to index. If the function Ssum::adjust_upper_limit_downwards(index) is used. If the function shift_plus_one() is called.
It checks, that letter does not give rise to poles (e.g. that the offset is a non-negative integer or a negative integer larger or equal to $c \ge o$ ).
It checks, that the in the Gamma functions are integers.

If one of the tests fails, the object is put into a zombie state.

If flag_expand_status == expand_status::check_for_poles, it assures that the Gamma functions in the numerator do not give rise to poles. The function shift_plus_one() is used.

If flag_expand_status == expand_status::expand_gamma_functions, the Gamma functions are expanded into Euler Zagier sums. This is done by setting the expansion_required flag in the ratio_of_tgamma class.

If flag_expand_status == expand_status::adjust_summation_index, we deal with sums of the form

$\mbox{} - \sum\limits_{i=1}^n \left( \begin{array}{c} n \\ i \\ \end{array} \right) \left( -1 \right)^i \frac{x^i}{(i+c)^m} S(i,...)$

with $c \ge 0$ . Here eval adjusts the offset. In the case the function shift_minus_one() is called. The function shift_minus_one() does not change the upper summation limit of the subsum (this avoids an infinite recursion).

If flag_expand_status == expand_status::do_hoelder_convolution, we deal with sums of the form

$\mbox{} - \sum\limits_{i=1}^n \left( \begin{array}{c} n \\ i \\ \end{array} \right) \left( -1 \right)^i \frac{x^i}{i^m} S(i,...)$

The S-sum is first brought to a standard form, which ensures that negative degrees are removed. The function Ssum::remove_negative_degrees() is used for that. We then check if is negative. In that case the function shift_plus_one() is called. We then can assume that is non-negative and that the subsum does not contain negative degrees. is then rewritten as

$S(i;m_1,...,m_k;x_1,...,x_k)$

$= S(N;m_1,...,m_k;x_1,...,x_k)$

$\mbox{} - S(N;m_2,...,m_k;x_2,...,x_k) \cdot \left( \sum\limits_{i_1=i+1}^N \frac{x_1^{i_1}}{i_1^{m_1}} \right)$

$+ S(N;m_3,...,m_k;x_3,...,x_k) \cdot \left( \sum\limits_{i_1=i+1}^N \sum\limits_{i_2=i_1+1}^N \frac{x_1^{i_1}}{i_1^{m_1}} \frac{x_2^{i_2}}{i_2^{m_2}} \right)$

$\mbox{} - ... + (-1)^k \cdot \left(\sum\limits_{i_1=i+1}^N \sum\limits_{i_2=i_1+1}^N ... \sum\limits_{i_k=i_{k-1}+1}^N \frac{x_1^{i_1}}{i_1^{m_1}} \frac{x_2^{i_2}}{i_2^{m_2}} ... \frac{x_k^{i_k}}{i_k^{m_k}} \right)$

Each term is a product of a Ssum at infinity and a sum of type "Csum". The Ssum at infinity is converted to a Zsum at infinity and expressed in terms of multiple polylogarithms. The evaluation of the Csum is done in its proper evaluation routine.

◆ eval_explicit()

ex eval_explicit ( ) const

virtual

Explicit evaluation

◆ eval_ncmul()

ex eval_ncmul ( const GiNaC::exvector & v ) const

overrideprotected

No automatic simplifications

◆ get_key()

unsigned get_key ( void ) const

virtual

The hash key is calculated from the hash_data.

◆ hash_data()

ex hash_data ( void ) const

virtual

The summation index is a redundant variable and does not influence the hash_data.

◆ set_expansion()

ex set_expansion ( void ) const

Sets the flag flag_expand_status to expand_status::expansion_required. The object is then automatically expanded up to the order specified in the member variable order.

◆ shift_minus_one()

ex shift_minus_one ( void ) const

This routine assumes sums of the form

$\mbox{} - \sum\limits_{i=1}^n \left( \begin{array}{c} n \\ i \\ \end{array} \right) \left( -1 \right)^i \frac{x^i}{(i+c)^m} S(i,...)$

and performs the substitution index -> index + 1.

If $m \le 0$ we use the binomial formula

$\mbox{} - \sum\limits_{i=1}^n \left( \begin{array}{c} n \\ i \\ \end{array} \right) \left( -1 \right)^i \frac{x^i}{(i+c)^m} S(i,...)$

$= \sum\limits_{j=0}^{-m} \left( \begin{array}{c} -m \\ j \\ \end{array} \right) c^{-m-j} \mbox{} (-1) \sum\limits_{i=1}^n \left( \begin{array}{c} n \\ i \\ \end{array} \right) \left( -1 \right)^i \frac{x^i}{i^{-j}} S(i,...)$

If the depth of the subsum is zero, we have

$\mbox{} - \sum\limits_{i=1}^n \left( \begin{array}{c} n \\ i \\ \end{array} \right) \left( -1 \right)^i \frac{x^i}{(i+c)^m} S(i)$

$= \mbox{} - \frac{1}{x} \frac{1}{n+1} Z(n-1) (-1) \sum\limits_{i=1}^{n+1} \left( \begin{array}{c} n+1 \\ i \\ \end{array} \right) \left( -1 \right)^i \frac{x^i}{(i+c-1)^m} i S(i) + \frac{1}{c^m} Z(n-1)$

If the depth of the subsum is not equal to zero, we have

$\mbox{} - \sum\limits_{i=1}^n \left( \begin{array}{c} n \\ i \\ \end{array} \right) \left( -1 \right)^i \frac{x^i}{(i+c)^m} S(i,...)$

$= \mbox{} - \frac{1}{x} \frac{1}{n+1} Z(n-1) (-1) \sum\limits_{i=1}^{n+1} \left( \begin{array}{c} n+1 \\ i \\ \end{array} \right) \left( -1 \right)^i \frac{x^i}{(i+c-1)^m} i S(i,...)$

$+ \frac{1}{x} \frac{1}{n+1} Z(n-1) (-1) \sum\limits_{i=1}^{n+1} \left( \begin{array}{c} n+1 \\ i \\ \end{array} \right) \left( -1 \right)^i \frac{x^i}{(i+c-1)^m} \frac{x_1^i}{i^{m_1-1}} S(i;m_2,...)$

This routine is called from eval/adjust_summation_index only for .

◆ shift_plus_one()

ex shift_plus_one ( void ) const

This routine performs the substitution index -> index - 1. The formula used is

$\mbox{} - \sum\limits_{i=1}^{n} \left( \begin{array}{c} n \\ i \\ \end{array} \right) \left( -1 \right)^i \frac{x^i}{(i+c)^m} \frac{\Gamma(i+a_1+b_1\varepsilon)}{\Gamma(i+c_1+d_1\varepsilon)} ... S(i+o,m_1,...,m_l,x_1,...,x_l)$

$= (-x) n (-1) \sum\limits_{i=1}^{n-1} \left( \begin{array}{c} n-1 \\ i \\ \end{array} \right) \left( -1 \right)^i \frac{x^i}{(i+c+1)^m} \frac{\Gamma(i+a_1+1+b_1\varepsilon)}{\Gamma(i+c_1+1+d_1\varepsilon)} ... \frac{1}{i+1} S(i+o+1,m_1,...,m_l,x_1,...,x_l)$