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transcendental_sum_type_B Class Reference

The class transcendental_sum_type_B may contain $i$ and $(n-i)$. More...

#include <transcendental_B.h>

List of all members.

Public Member Functions

 transcendental_sum_type_B (const GiNaC::ex &nn, const GiNaC::ex &i, const GiNaC::ex &l, const GiNaC::ex &lr, const GiNaC::ex &v, const GiNaC::ex &vr, const GiNaC::ex &ss, const GiNaC::ex &ssr, const GiNaC::ex &eps, int o, int f)
void archive (GiNaC::archive_node &node) const
void read_archive (const GiNaC::archive_node &node, GiNaC::lst &sym_lst)
unsigned return_type (void) const
void print (const GiNaC::print_context &c, unsigned level=0) const
unsigned precedence (void) const
GiNaC::ex eval (int level=0) const
GiNaC::ex subs (const GiNaC::exmap &m, unsigned options=0) const
virtual GiNaC::ex eval_explicit (int level=0) const
GiNaC::ex set_expansion (void) const
GiNaC::ex distribute_over_subsum (void) const
GiNaC::ex distribute_over_letter (void) const
GiNaC::ex distribute_over_subsum_rev (void) const
GiNaC::ex distribute_over_letter_rev (void) const
GiNaC::ex shift_plus_one (void) const
GiNaC::ex shift_plus_one_rev (void) const

Protected Member Functions

GiNaC::ex eval_ncmul (const GiNaC::exvector &v) const
GiNaC::ex derivative (const GiNaC::symbol &s) const
unsigned calchash (void) const

Protected Attributes

GiNaC::ex n
GiNaC::ex index
GiNaC::ex letter
GiNaC::ex letter_rev
GiNaC::ex lst_of_gammas
GiNaC::ex lst_of_gammas_rev
GiNaC::ex subsum
GiNaC::ex subsum_rev
GiNaC::ex expansion_parameter
int order
int flag_expand_status

Detailed Description

The class transcendental_sum_type_B may contain $i$ and $(n-i)$.

The definition is

\[ \frac{\Gamma(1+d_1\varepsilon)}{\Gamma(1+b_1\varepsilon)} ... \frac{\Gamma(1+d_k\varepsilon)}{\Gamma(1+b_k\varepsilon)} \frac{\Gamma(1+d_1'\varepsilon)}{\Gamma(1+b_1'\varepsilon)} ... \frac{\Gamma(1+d_{k'}'\varepsilon)}{\Gamma(1+b_{k'}'\varepsilon)} \]

\[ \sum\limits_{i=1}^{n-1} \frac{x^i}{(i+c)^m} \frac{\Gamma(i+a_1+b_1\varepsilon)}{\Gamma(i+c_1+d_1\varepsilon)} \frac{\Gamma(i+a_2+b_2\varepsilon)}{\Gamma(i+c_2+d_2\varepsilon)} ... \frac{\Gamma(i+a_k+b_k\varepsilon)}{\Gamma(i+c_k+d_k\varepsilon)} Z(i+o-1,m_1,...,m_l,x_1,...,x_l) \]

\[ \cdot \frac{y^{n-i}}{(n-i+c')^{m'}} \frac{\Gamma(n-i+a_1'+b_1'\varepsilon)}{\Gamma(n-i+c_1'+d_1'\varepsilon)} \frac{\Gamma(n-i+a_2'+b_2'\varepsilon)}{\Gamma(n-i+c_2'+d_2'\varepsilon)} ... \frac{\Gamma(n-i+a_{k'}'+b_{k'}'\varepsilon)}{\Gamma(n-i+c_{k'}'+d_{k'}'\varepsilon)} Z(n-i+o'-1,m_1',...,m_{l'}',x_1',...,x_{l'}') \]

Here, $c$, $c'$, $o$, $o'$ and all $a_j$, $a_j'$, $c_j$ and $c_j'$ are integers. In addition $c$ is non-negative or $ c \ge -l+o $. Simmilar, $c'$ is non-negative or $ c' \ge -l'+o' $.

Note that the upper summation limit is $(n-1)$. The upper summation limit should not be infinity.

Member Function Documentation

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_B to a canonical form, so that afterwards letter only contains a basic_letter.

ex distribute_over_letter_rev (  void   )  const
 

letter_rev 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_B to a canonical form, so that afterwards letter_rev only contains a basic_letter.

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_B to a canonical form, so that afterwards subsum only contains a Zsum.

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 Zsum 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 Ssum and the difference of its index with the index member is an integer, it is converted to an Zsum and stays inside the subsum. Otherwise it is taken out of the sum.

ex distribute_over_subsum_rev (  void   )  const
 

subsum_rev 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_B to a canonical form, so that afterwards subsum_rev only contains a Zsum.

The algorithm is based on the following steps:

  • If an object is of type basic_letter and the difference of its index with $(n-i)$ is an integer, it is combined with the letter_rev 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 $(n-i)$ is an integer, it is combined with the lst_of_gammas_rev member. Otherwise it is taken out of the sum.

  • If an object is of type Zsum and the difference of its index with $(n-i)$ is an integer, it is stays inside the subsum_rev. Otherwise it is taken out of the sum.

  • If an object is of type Ssum and the difference of its index with $(n-i)$ is an integer, it is converted to an Zsum and stays inside the subsum_rev. Otherwise it is taken out of the sum.

ex eval (  int  level = 0  )  const
 

Simplifications, which are always performed are:

  • If subsum is equal to 1, the subsum is replaced with Z(i-1).

  • If subsum_rev is equal to 1, the subsum_rev is replaced with Z(n-i-1).

  • If subsum is not of type Zsum, the routine distribute_over_subsum is called.

  • If the difference of the upper summation index of the Zsum (in subsum) with the index member is not an integer, the Zsum is taken out of the sum. The subsum is replaced with Z(i-1).

  • If subsum_rev is not of type Zsum, the routine distribute_over_subsum_rev is called.

  • If the difference of the upper summation index of the Zsum (in subsum_rev) with $(n-i)$ is not an integer, the Zsum is taken out of the sum. The subsum_rev is replaced with Z(n-i-1).

  • 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 (in letter) with the index member is not zero, the index in basic_letter is adjusted.

  • If letter_rev is not of type basic_letter, the routine distribute_over_letter_rev is called.

  • If the difference of the index of the basic_letter (in letter_rev) with $(n-i)$ 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 the difference of the index of lst_of_gammas_rev with $(n-i)$ is not zero, the index in lst_of_gammas_rev is adjusted.

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

  • It checks, that letter and letter_rev do not give rise to poles (e.g. that the offset $c$ is a non-negative integer or a negative integer larger or equal to $c \ge -l+o $ and that the offset $c'$ is a non-negative integer or a negative integer larger or equal to $c' \ge -l'+o' $).

  • It checks, that the $a_j$ 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 functions shift_plus_one() and shift_plus_one_rev() are 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::do_partial_fractioning, the sum is of the form

\[ \sum\limits_{i=1}^{n-1} \frac{x^i}{(i+c)^m} Z(i-1,m_1,...) \frac{y^{n-i}}{(n-i+c')^{m'}} Z(n-i-1,m_1',....) \]

For $m=0$ we have

\[ x^n \sum\limits_{i=1}^{n-1} \frac{\left(\frac{y}{x}\right)^i}{(i+c')^{m'}} Z(i-1,m_1',...) Z(n-i-1,m_1,...) \]

For $m'=0$ we have

\[ y^n \sum\limits_{i=1}^{n-1} \frac{\left(\frac{x}{y}\right)^i}{(i+c)^{m}} Z(i-1,m_1,...) Z(n-i-1,m_1',...) \]

For $m'<0$ we use

\[ (n-i+c') = (n+c') - i \]

For $m<0$ we use

\[ (i+c) = (n+c) - (n-i) \]

For $m>0$ and $m'>0$ we have

\[ \sum\limits_{i=1}^{n-1} \frac{x^i}{(i+c)^m} Z(i-1,m_1,...) \frac{y^{n-i}}{(n-i+c')^{m'}} Z(n-i-1,m_1',....) \]

\[ = \frac{1}{n+c+c'} \sum\limits_{i=1}^{n-1} \frac{x^i}{(i+c)^{m-1}} Z(i-1,...) \frac{y^{n-i}}{(n-i+c')^{m'}} Z(n-i-1,....) \]

\[ \mbox{} + \frac{1}{n+c+c'} \sum\limits_{i=1}^{n-1} \frac{x^i}{(i+c)^{m}} Z(i-1,...) \frac{y^{n-i}}{(n-i+c')^{m'-1}} Z(n-i-1,....) \]

If flag_expand_status == expand_status::do_outermost_sum, the sum is of the form

\[ \sum\limits_{i=1}^{n-1} \frac{x^i}{(i+c)^m} Z(i-1,m_1,...) Z(n-i-1,m_1',....) \]

If the depth of $Z(n-i-1,m_1',....)$ is zero, we have a sum of type A with upper summation index $(n-1)$:

\[ \sum\limits_{i=1}^{n-1} \frac{x^i}{(i+c)^m} Z(i-1,m_1,...) \]

Otherwise we have the recursion

\[ \sum\limits_{i=1}^{n-1} \frac{x^i}{(i+c)^m} Z(i-1,m_1,...) Z(n-i-1,m_1',....) \]

\[ = \sum\limits_{j=1}^{n-1} \left[ \sum\limits_{i=1}^{j-1} \frac{x^i}{(i+c)^m} Z(i-1,m_1,...) \frac{{x_1'}^{j-i}}{(j-i)^{m_1'}} Z(j-i-1,m_2',....) \right] \]

ex eval_explicit (  int  level = 0  )  const [virtual]
 

Explicit evaluation

ex eval_ncmul (  const GiNaC::exvector &  v  )  const [protected]
 

No automatic simplifications

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.

ex shift_plus_one (  void   )  const
 

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

\[ \sum\limits_{i=1}^{n-1} \frac{x^i}{(i+c)^m} \frac{\Gamma(i+a_1+b_1\varepsilon)}{\Gamma(i+c_1+d_1\varepsilon)} ... Z(i-1+o,m_1,...,m_l,x_1,...,x_l) \cdot \frac{y^{n-i}}{(n-i+c')^{m'}} \frac{\Gamma(n-i+a_1'+b_1'\varepsilon)}{\Gamma(n-i+c_1'+d_1'\varepsilon)} ... Z(n-i-1+o',m_1',...,m_{l'}',x_1',...,x_{l'}') \]

\[ = x \sum\limits_{i=1}^{n-2} \frac{x^i}{(i+c+1)^m} \frac{\Gamma(i+a_1+1+b_1\varepsilon)}{\Gamma(i+c_1+1+d_1\varepsilon)} ... Z(i+o,m_1,...,m_l,x_1,...,x_l) \cdot \frac{y^{n-1-i}}{(n-1-i+c')^{m'}} \frac{\Gamma(n-1-i+a_1'+b_1'\varepsilon)}{\Gamma(n-1-i+c_1'+d_1'\varepsilon)} ... Z(n-1-i-1+o',m_1',...,m_{l'}',x_1',...,x_{l'}') \]

\[ + \frac{x}{(c+1)^m} \frac{\Gamma(a_1+1+b_1\varepsilon)}{\Gamma(c_1+1+d_1\varepsilon)} ... Z(o,m_1,...,m_l,x_1,...,x_l) \cdot \frac{y^{n-1}}{(n-1+c')^{m'}} \frac{\Gamma(n-1+a_1'+b_1'\varepsilon)}{\Gamma(n-1+c_1'+d_1'\varepsilon)} ... Z(n-2+o',m_1',...,m_{l'}',x_1',...,x_{l'}') Z(n-2) \]

This routine is called from eval/expansion_required, and eval/check_for_poles.

ex shift_plus_one_rev (  void   )  const
 

This routine takes out the term at index = n - 1. The formula used is

\[ \sum\limits_{i=1}^{n-1} \frac{x^i}{(i+c)^m} \frac{\Gamma(i+a_1+b_1\varepsilon)}{\Gamma(i+c_1+d_1\varepsilon)} ... Z(i-1+o,m_1,...,m_l,x_1,...,x_l) \cdot \frac{y^{n-i}}{(n-i+c')^{m'}} \frac{\Gamma(n-i+a_1'+b_1'\varepsilon)}{\Gamma(n-i+c_1'+d_1'\varepsilon)} ... Z(n-i-1+o',m_1',...,m_{l'}',x_1',...,x_{l'}') \]

\[ = y \sum\limits_{i=1}^{n-2} \frac{x^i}{(i+c)^m} \frac{\Gamma(i+a_1+b_1\varepsilon)}{\Gamma(i+c_1+d_1\varepsilon)} ... Z(i-1+o,m_1,...,m_l,x_1,...,x_l) \cdot \frac{y^{n-1-i}}{(n-1-i+c'+1)^{m'}} \frac{\Gamma(n-1-i+a_1'+1+b_1'\varepsilon)}{\Gamma(n-1-i+c_1'+1+d_1'\varepsilon)} ... Z(n-1-i+o',m_1',...,m_{l'}',x_1',...,x_{l'}') \]

\[ + \frac{x^{n-1}}{(n-1+c)^m} \frac{\Gamma(n-1+a_1+b_1\varepsilon)}{\Gamma(n-1+c_1+d_1\varepsilon)} ... Z(n-2+o,m_1,...,m_l,x_1,...,x_l) \cdot \frac{y}{(1+c')^{m'}} \frac{\Gamma(a_1'+1+b_1'\varepsilon)}{\Gamma(c_1'+1+d_1'\varepsilon)} ... Z(o',m_1',...,m_{l'}',x_1',...,x_{l'}') Z(n-2) \]

This routine is called from eval/expansion_required, and eval/check_for_poles.

The documentation for this class was generated from the following files:
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