![\[ \sum\limits_{i_1=n+1}^N \sum\limits_{i_2=i_1+1}^N ... \sum\limits_{i_k=i_{k-1}+1}^N \frac{x_1^{i_1}}{i_1^{m_1}} \frac{x_2^{i_2}}{i_2^{m_2}} ... \frac{x_k^{i_k}}{i_k^{m_k}} \]](form_24.png)
#include <Bsum.h>
Public Member Functions | |
| Bsum (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) |
| 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 | convert_to_Ssum_exvector (const GiNaC::exvector &Z1) const |
| virtual GiNaC::ex | expand_members (int level=0) const |
| GiNaC::ex | get_index (void) const |
| GiNaC::ex | get_letter_list (void) const |
| unsigned | get_depth (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 | letter_list |
Friends | |
| GiNaC::ex | convert_Bsum_to_Ssum (const GiNaC::ex &C) |
| GiNaC::ex | convert_Bsum_to_Zsum (const GiNaC::ex &C) |
A Bsum is defined by
This is equivalent to
![\[ \left( x_k^+ \right)^{m_k} \left( x_{k-1}^+ \right)^{m_{k-1}} ... \left( x_1^+ \right)^{m_1} \frac{x_k}{1-x_k} \frac{x_{k-1}x_k}{1-x_{k-1}x_k} ... \frac{x_1 ... x_k}{1-x_1 ... x_k} \left( x_1 ... x_k \right)^n \]](form_25.png)
up to terms of the form
![\[ \left( x^+ \right)^m \frac{x}{1-x} x^N = \sum\limits_{i=N+1}^\infty \frac{x^i}{i^m} \]](form_26.png)
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The basic object is
We use
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No automatic simplifications |
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No automatic simplifications |
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Returns the depth. |
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Returns the upper summation limit. |
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Returns the letter_list. |
1.3.7
![\[ \left( \sum\limits_{j_1=1}^{n'} \sum\limits_{j_2=1}^{j_1} ... \sum\limits_{n=1}^{j_{l-1}} \frac{y_1^{j_1}}{j_1^{g_1}} ... \frac{y_l^{n}}{n^{g_l}} \right) \sum\limits_{i_1=n+1}^{N} ... \sum\limits_{i_k = i_{k-1}+1}^{N} \frac{x_1^{i_1}}{i_1^{m_1}} ... \frac{x_k^{i_k}}{i_k^{m_k}} \]](form_19.png)
![\[ \sum\limits_{i_1=n+1}^{N} ... \sum\limits_{i_k = i_{k-1}+1}^{N} \frac{x_1^{i_1}}{i_1^{m_1}} ... \frac{x_k^{i_k}}{i_k^{m_k}} \]](form_20.png)
![\[ = (-1)^k S(n;m_1,...m_k;x_1,...,x_k) - (-1)^k S(N;m_1,...m_k;x_1,...,x_k) \]](form_21.png)
![\[ + (-1)^k S(N;m_2,...m_k;x_2,...,x_k) \sum\limits_{i_1=n+1}^{N} \frac{x_1^{i_1}}{i_1^{m_1}} ... \]](form_22.png)
![\[ + (-1)^k S(N;m_k;x_k) \sum\limits_{i_1=n+1}^{N} ... \sum\limits_{i_{k-1} = i_{k-2}+1}^{N} \frac{x_1^{i_1}}{i_1^{m_1}} ... \frac{x_{k-1}^{i_{k-1}}}{i_{k-1}^{m_{k-1}}} \]](form_23.png)