Euler_Zagier_sum Class Reference

A Euler-Zagier sum is a special case of a Zsum. More...

#include <Euler_Zagier_sum.h>

Inheritance diagram for Euler_Zagier_sum:


Public Member Functions
	Euler_Zagier_sum (const GiNaC::ex &nc)
	Euler_Zagier_sum (const GiNaC::ex &nc, const GiNaC::ex &llc)
void	archive (GiNaC::archive_node &node) const
void	read_archive (const GiNaC::archive_node &node, GiNaC::lst &sym_lst)
GiNaC::return_type_t	return_type_tinfo () const
void	print (const GiNaC::print_context &c, unsigned level=0) const
GiNaC::ex	eval (int level=0) const
GiNaC::ex	convert_to_Ssum_exvector (const GiNaC::exvector &Z0, const GiNaC::exvector &Z1) const
GiNaC::ex	shuffle_exvector (const GiNaC::exvector &Z0, const GiNaC::exvector &Z1, const GiNaC::exvector &Z2) const
GiNaC::ex	set_index (const GiNaC::ex &i) const
GiNaC::ex	shift_plus_one (void) const
GiNaC::ex	shift_minus_one (void) const
GiNaC::ex	adjust_upper_limit_downwards (const GiNaC::ex &i) const
GiNaC::ex	adjust_upper_limit_upwards (const GiNaC::ex &i) const
GiNaC::ex	adjust_upper_limit_plus_one (void) const
GiNaC::ex	remove_first_letter (void) const
GiNaC::ex	remove_first_letter (const GiNaC::ex &nc) const

Detailed Description

A Euler-Zagier sum is a special case of a Zsum.

Euler-Zagier sums are recursively defined by

$Z_{m_1,...,m_k}(n) = \sum\limits_{i=1}^n \frac{1}{i^{m_1}} Z_{m_2,...,m_k}(i-1)$

with

$Z(n) = 1$

for $n \ge 0$ .

Member Function Documentation

ex adjust_upper_limit_downwards ( const GiNaC::ex & i ) const [virtual]

Adjusts the upper summation limit $n$ down to $i$ , e.g.
$Z(n;m_1,...;1,...) = Z(i;m_1,...;1,...) + \sum\limits_{j=1}^{n-i} \frac{1}{(i+j)^{m_1}} Z(i-1+j;m_2,...;1,...)$
with $n > i $ .
For the empty sum we have
$Z(n) = Z(i)$

This routine assumes $i \ge 0$ .
Reimplemented from Zsum.

ex adjust_upper_limit_plus_one ( void ) const [virtual]

Adjusts the upper summation limit $n$ to $n+1$ , e.g.
$Z(n;m_1,...;1,...) = Z(n+1;m_1,...;1,...) \mbox{} - \frac{1}{(n+1)^{m_1}} Z(n;m_2,...;1,...).$

Reimplemented from Zsum.

ex adjust_upper_limit_upwards ( const GiNaC::ex & i ) const [virtual]

Adjusts the upper summation limit $n$ up to $i$ , e.g.
$Z(n;m_1,...;1,...) = Z(i;m_1,...;1,...) \mbox{} - \sum\limits_{j=0}^{i-n-1} \frac{1}{(i-j)^{m_1}} Z(i-j-1;m_2,...;1,...)$
with $n < i $ .
For the empty sum we have
$Z(n) = Z(i)$

This routine assumes $n \ge 0$ .
This routine might give rise to poles for some values of $i$ and should be used with care.
Reimplemented from Zsum.

ex convert_to_Ssum_exvector ( const GiNaC::exvector & Z0,

const GiNaC::exvector & Z1

) const [virtual]

A more efficient version for the conversion to harmonic sums, based on exvector (e.g. std::vector<GiNaC::ex> ). Z0 contains the result. Z1 is reversed order, so that we can use pop_back.
Reimplemented from Zsum.

ex eval ( int level = 0 ) const

The simplifications are done in the following order:

If the upper summation limit is equal to infinity, we have a multiple zeta value.

If the upper summation index is an integer, perform the sum explicitly.

Reimplemented from Zsum.
Reimplemented in multiple_zeta_value.

ex remove_first_letter ( const GiNaC::ex & nc ) const [virtual]

Returns a Euler_Zagier_sum with the first letter removed from the letter_list and the upper summation index set to nc.
Reimplemented from Zsum.

ex remove_first_letter ( void ) const [virtual]

Returns a Euler_Zagier_sum with the first letter removed from the letter_list.
Reimplemented from Zsum.

ex set_index ( const GiNaC::ex & i ) const [virtual]

Sets the upper summation index to $i$
Reimplemented from Zsum.

ex shift_minus_one ( void ) const [virtual]

Returns $Z(n-1,m_1,...,1,...)$
Reimplemented from Zsum.

ex shift_plus_one ( void ) const [virtual]

Returns $Z(n+1,m_1,...,1,...)$
Reimplemented from Zsum.

ex shuffle_exvector ( const GiNaC::exvector & Z0,

const GiNaC::exvector & Z1,

const GiNaC::exvector & Z2

) const [virtual]

A more efficient version for the multiplication of Euler_Zagier_sums, based on exvector (e.g. std::vector<GiNaC::ex> ). Z0 contains the result. Z1 and Z2 are in reversed order, so that we can use pop_back.
Reimplemented from Zsum.

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

Generated on Wed Jun 10 22:59:10 2009 for Nestedsums library by

1.3.7

Euler_Zagier_sum Class Reference

Public Member Functions

Detailed Description

Member Function Documentation