‘Mathematical’ Cracks Versus Artificial Slits: Implications in the Determination of Fracture Toughness

Abstract

An analytic solution is introduced for the stress field developed in a circular finite disc weakened by a central slit of arbitrary ratio of its edges and slightly rounded corners. The disc is loaded by radial pressure applied along two finite arcs of its periphery, anti-symmetric with respect to the disc’s center. The motive of the study is to consider the stress field in a disc with a mechanically machined slit (finite distance between the two lips) in juxtaposition to the respective field in the same disc with a ‘mathematical’ crack (zero distance between lips), which is the configuration adopted in case the fracture toughness of brittle materials is determined according to the standardized cracked Brazilian-disc test. The solution is obtained using Muskhelishvili’s complex potentials’ technique adopting a suitable conformal mapping function found, also, in Savin’s milestone book. For the task to be accomplished, an auxiliary problem is first solved, namely, the infinite plate with a rectangular slit (in case the resultant force on the slit is zero and also the stresses and rotations at infinity are zero), by mapping conformally the area outside the slit onto the mathematical plane with a unit hole. The formulae obtained for the complex potentials permit the analytic exploration of the stress field along some loci of crucial practical importance. The influence of the slit’s width on the local stress amplification and also on the stress concentration around the crown of the slit is quantitatively described. In addition, the role of the load-application mode (compression along the slit’s longitudinal symmetry axis and tension normal to it) is explored. Results indicate that the two configurations are not equivalent in terms of the stress concentration factor. In addition, depending on the combination of the slit’s width and the load-application mode, the point where the normal stress along the slit’s boundary is maximized ‘oscillates’ between the central point of the slit’s short edge (intersection of the slit’s longitudinal axis with its perimeter) and the slit’s corners.

Appendix

Principal parts of the following functions in brackets, appearing in Eqs. (28), (29), (39) and (40)

$$ \begin{aligned} G_{4n + 1}^{\left( \infty \right)} \left( \zeta \right) & : = \mathop {P.P.}\limits_{\zeta = \,\,\infty } \left[ {\left( {\zeta + \frac{{c_{1} }}{\zeta } + \frac{{c_{2} }}{{\zeta^{3} }} + \frac{{c_{3} }}{{\zeta^{5} }} + \frac{{c_{4} }}{{\zeta^{7} }} + \frac{{c_{5} }}{{\zeta^{9} }} + \frac{{c_{6} }}{{\zeta^{11} }}} \right)^{4n + 1} } \right]{\kern 1pt} \\ & = \sum\limits_{{k_{1} = 0}}^{4n + 1} { \cdot \sum\limits_{{k_{2} = 0}}^{{4n + 1 - k_{1} }} { \cdot \sum\limits_{{k_{3} = 0}}^{{4n + 1 - \,\,\left( {k_{1} + k_{2} } \right)}} { \cdot \sum\limits_{{k_{4} = 0}}^{{4n + 1 - \,\,\left( {k_{1} + k_{2} + k_{3} } \right)}} { \cdot \sum\limits_{{k_{5} = 0}}^{{4n + 1 - \,\,\left( {k_{1} + k_{2} + k_{3} + k_{4} } \right)}} { \cdot \sum\limits_{{k_{6} = 0}}^{{\frac{4n + 1}{2}\,\, - \,\,\left( {6k_{1} + 5k_{2} + 4k_{3} + 3k_{4} + 2k_{5} } \right)}} {} } } } } } \\ & \quad \frac{{\left( {4n + 1} \right)!\,\,c_{6}^{{k_{1} }} c_{5}^{{k_{2} }} c_{4}^{{k_{3} }} c_{3}^{{k_{4} }} c_{2}^{{k_{5} }} c_{1}^{{k_{6} }} }}{{k_{1} !k_{2} !k_{3} !k_{4} !k_{5} !k_{6} !\left( {4n + 1 - k_{1} - k_{2} - k_{3} - k_{4} - k_{5} - k_{6} } \right)!\,}}\zeta^{{\,4n + 1\, - \,2\,\left( {6k_{1} + 5k_{2} + 4k_{3} + 3k_{4} + 2k_{5} + k_{6} } \right)}} \\ \end{aligned} $$

$$ \begin{aligned} G_{4n}^{\left( 0 \right)} \left( \zeta \right): = & \mathop {P.P.}\limits_{\zeta = \,\,0} \left[ {\left( {\zeta + \frac{{c_{1} }}{\zeta } + \frac{{c_{2} }}{{\zeta^{3} }} + \frac{{c_{3} }}{{\zeta^{5} }} + \frac{{c_{4} }}{{\zeta^{7} }} + \frac{{c_{5} }}{{\zeta^{9} }} + \frac{{c_{6} }}{{\zeta^{11} }}} \right)\left( {\frac{1}{\zeta } + c_{1} \zeta + c_{2} \zeta^{3} + c_{3} \zeta^{5} + c_{4} \zeta^{7} + c_{5} \zeta^{9} + c_{6} \zeta^{11} } \right)^{4n} } \right]{\kern 1pt} = \\ & \quad \sum\limits_{{k_{1} = 0}}^{4n} { \cdot \sum\limits_{{k_{2} = 0}}^{{4n - k_{1} }} { \cdot \sum\limits_{{k_{3} = 0}}^{{4n - \,\,\left( {k_{1} + k_{2} } \right)}} { \cdot \sum\limits_{{k_{4} = 0}}^{{4n - \,\,\left( {k_{1} + k_{2} + k_{3} } \right)}} { \cdot \sum\limits_{{k_{5} = 0}}^{{4n - \,\,\left( {k_{1} + k_{2} + k_{3} + k_{4} } \right)}} { \cdot \sum\limits_{{k_{6} = 0}}^{{2n - 1\, - \,\,\left( {6k_{1} + 5k_{2} + 4k_{3} + 3k_{4} + 2k_{5} } \right)}} {} } } } } } \\ & \quad \frac{{\left( {4n} \right)!\,\,c_{6}^{{k_{1} }} c_{5}^{{k_{2} }} c_{4}^{{k_{3} }} c_{3}^{{k_{4} }} c_{2}^{{k_{5} }} c_{1}^{{k_{6} }} }}{{k_{1} !k_{2} !k_{3} !k_{4} !k_{5} !k_{6} !\left( {4n - k_{1} - k_{2} - k_{3} - k_{4} - k_{5} - k_{6} } \right)!\,}}\zeta^{{ - \,4n + 1 + \,2\,\left( {6k_{1} + 5k_{2} + 4k_{3} + 3k_{4} + 2k_{5} + k_{6} } \right)}} + \\ & \quad \sum\limits_{{k_{1} = 0}}^{4n} { \cdot \sum\limits_{{k_{2} = 0}}^{{4n - k_{1} }} { \cdot \sum\limits_{{k_{3} = 0}}^{{4n - \,\,\left( {k_{1} + k_{2} } \right)}} { \cdot \sum\limits_{{k_{4} = 0}}^{{4n - \,\,\left( {k_{1} + k_{2} + k_{3} } \right)}} { \cdot \sum\limits_{{k_{5} = 0}}^{{4n - \,\,\left( {k_{1} + k_{2} + k_{3} + k_{4} } \right)}} { \cdot \sum\limits_{{k_{6} = 0}}^{{2n - \,\,\left( {6k_{1} + 5k_{2} + 4k_{3} + 3k_{4} + 2k_{5} } \right)}} {} } } } } } \\ & \quad \frac{{\left( {4n} \right)!\,\,c_{6}^{{k_{1} }} c_{5}^{{k_{2} }} c_{4}^{{k_{3} }} c_{3}^{{k_{4} }} c_{2}^{{k_{5} }} c_{1}^{{k_{6} + 1}} }}{{k_{1} !k_{2} !k_{3} !k_{4} !k_{5} !k_{6} !\left( {4n - k_{1} - k_{2} - k_{3} - k_{4} - k_{5} - k_{6} } \right)!\,}}\zeta^{{\, - \,\,\left( {4n + 1} \right) + \,2\,\left( {6k_{1} + 5k_{2} + 4k_{3} + 3k_{4} + 2k_{5} + k_{6} } \right)}} + \\ & \quad \sum\limits_{{k_{1} = 0}}^{4n} { \cdot \sum\limits_{{k_{2} = 0}}^{{4n - k_{1} }} { \cdot \sum\limits_{{k_{3} = 0}}^{{4n - \,\,\left( {k_{1} + k_{2} } \right)}} { \cdot \sum\limits_{{k_{4} = 0}}^{{4n - \,\,\left( {k_{1} + k_{2} + k_{3} } \right)}} { \cdot \sum\limits_{{k_{5} = 0}}^{{4n - \,\,\left( {k_{1} + k_{2} + k_{3} + k_{4} } \right)}} { \cdot \sum\limits_{{k_{6} = 0}}^{{2n + 1 - \,\,\left( {6k_{1} + 5k_{2} + 4k_{3} + 3k_{4} + 2k_{5} } \right)}} {} } } } } } \\ & \quad \frac{{\left( {4n} \right)!\,\,c_{6}^{{k_{1} }} c_{5}^{{k_{2} }} c_{4}^{{k_{3} }} c_{3}^{{k_{4} }} c_{2}^{{k_{5} + 1}} c_{1}^{{k_{6} }} }}{{k_{1} !k_{2} !k_{3} !k_{4} !k_{5} !k_{6} !\left( {4n - k_{1} - k_{2} - k_{3} - k_{4} - k_{5} - k_{6} } \right)!\,}}\zeta^{{\, - \,\,\left( {4n + 3} \right) + \,2\,\left( {6k_{1} + 5k_{2} + 4k_{3} + 3k_{4} + 2k_{5} + k_{6} } \right)}} + \\ & \quad \sum\limits_{{k_{1} = 0}}^{4n} { \cdot \sum\limits_{{k_{2} = 0}}^{{4n - k_{1} }} { \cdot \sum\limits_{{k_{3} = 0}}^{{4n - \,\,\left( {k_{1} + k_{2} } \right)}} { \cdot \sum\limits_{{k_{4} = 0}}^{{4n - \,\,\left( {k_{1} + k_{2} + k_{3} } \right)}} { \cdot \sum\limits_{{k_{5} = 0}}^{{4n - \,\,\left( {k_{1} + k_{2} + k_{3} + k_{4} } \right)}} { \cdot \sum\limits_{{k_{6} = 0}}^{{2\,\left( {n + 1} \right)\, - \,\,\left( {6k_{1} + 5k_{2} + 4k_{3} + 3k_{4} + 2k_{5} } \right)}} {} } } } } } \\ & \quad \frac{{\left( {4n} \right)!\,\,c_{6}^{{k_{1} }} c_{5}^{{k_{2} }} c_{4}^{{k_{3} }} c_{3}^{{k_{4} + 1}} c_{2}^{{k_{5} }} c_{1}^{{k_{6} }} }}{{k_{1} !k_{2} !k_{3} !k_{4} !k_{5} !k_{6} !\left( {4n - k_{1} - k_{2} - k_{3} - k_{4} - k_{5} - k_{6} } \right)!\,}}\zeta^{{\, - \,\,\left( {4n + 5} \right) + \,2\,\left( {6k_{1} + 5k_{2} + 4k_{3} + 3k_{4} + 2k_{5} + k_{6} } \right)}} + \\ & \quad \sum\limits_{{k_{1} = 0}}^{4n} { \cdot \sum\limits_{{k_{2} = 0}}^{{4n - k_{1} }} { \cdot \sum\limits_{{k_{3} = 0}}^{{4n - \,\,\left( {k_{1} + k_{2} } \right)}} { \cdot \sum\limits_{{k_{4} = 0}}^{{4n - \,\,\left( {k_{1} + k_{2} + k_{3} } \right)}} { \cdot \sum\limits_{{k_{5} = 0}}^{{4n - \,\,\left( {k_{1} + k_{2} + k_{3} + k_{4} } \right)}} { \cdot \sum\limits_{{k_{6} = 0}}^{{2n + 3\, - \,\,\left( {6k_{1} + 5k_{2} + 4k_{3} + 3k_{4} + 2k_{5} } \right)}} {} } } } } } \\ & \quad \frac{{\left( {4n} \right)!\,\,c_{6}^{{k_{1} }} c_{5}^{{k_{2} }} c_{4}^{{k_{3} + 1}} c_{3}^{{k_{4} }} c_{2}^{{k_{5} }} c_{1}^{{k_{6} }} }}{{k_{1} !k_{2} !k_{3} !k_{4} !k_{5} !k_{6} !\left( {4n - k_{1} - k_{2} - k_{3} - k_{4} - k_{5} - k_{6} } \right)!\,}}\zeta^{{\, - \,\,\left( {4n + 7} \right) + \,2\,\left( {6k_{1} + 5k_{2} + 4k_{3} + 3k_{4} + 2k_{5} + k_{6} } \right)}} \\ & + \sum\limits_{{k_{1} = 0}}^{4n} { \cdot \sum\limits_{{k_{2} = 0}}^{{4n - k_{1} }} { \cdot \sum\limits_{{k_{3} = 0}}^{{4n - \,\,\left( {k_{1} + k_{2} } \right)}} { \cdot \sum\limits_{{k_{4} = 0}}^{{4n - \,\,\left( {k_{1} + k_{2} + k_{3} } \right)}} { \cdot \sum\limits_{{k_{5} = 0}}^{{4n - \,\,\left( {k_{1} + k_{2} + k_{3} + k_{4} } \right)}} { \cdot \sum\limits_{{k_{6} = 0}}^{{2\,\left( {n + 2} \right)\, - \,\,\left( {6k_{1} + 5k_{2} + 4k_{3} + 3k_{4} + 2k_{5} } \right)}} {} } } } } } \\ & \quad \frac{{\left( {4n} \right)!\,\,c_{6}^{{k_{1} }} c_{5}^{{k_{2} + 1}} c_{4}^{{k_{3} }} c_{3}^{{k_{4} }} c_{2}^{{k_{5} }} c_{1}^{{k_{6} }} }}{{k_{1} !k_{2} !k_{3} !k_{4} !k_{5} !k_{6} !\left( {4n - k_{1} - k_{2} - k_{3} - k_{4} - k_{5} - k_{6} } \right)!\,}}\zeta^{{\, - \,\,\left( {4n + 9} \right) + \,2\,\left( {6k_{1} + 5k_{2} + 4k_{3} + 3k_{4} + 2k_{5} + k_{6} } \right)}} + \\ & \quad \sum\limits_{{k_{1} = 0}}^{4n} { \cdot \sum\limits_{{k_{2} = 0}}^{{4n - k_{1} }} { \cdot \sum\limits_{{k_{3} = 0}}^{{4n - \,\,\left( {k_{1} + k_{2} } \right)}} { \cdot \sum\limits_{{k_{4} = 0}}^{{4n - \,\,\left( {k_{1} + k_{2} + k_{3} } \right)}} { \cdot \sum\limits_{{k_{5} = 0}}^{{4n - \,\,\left( {k_{1} + k_{2} + k_{3} + k_{4} } \right)}} { \cdot \sum\limits_{{k_{6} = 0}}^{{2\,n + 5\, - \,\,\left( {6k_{1} + 5k_{2} + 4k_{3} + 3k_{4} + 2k_{5} } \right)}} {} } } } } } \\ & \quad \frac{{\left( {4n} \right)!\,\,c_{6}^{{k_{1} + 1}} c_{5}^{{k_{2} }} c_{4}^{{k_{3} }} c_{3}^{{k_{4} }} c_{2}^{{k_{5} }} c_{1}^{{k_{6} }} }}{{k_{1} !k_{2} !k_{3} !k_{4} !k_{5} !k_{6} !\left( {4n - k_{1} - k_{2} - k_{3} - k_{4} - k_{5} - k_{6} } \right)!\,}}\zeta^{{\, - \,\,\left( {4n + 11} \right) + \,2\,\left( {6k_{1} + 5k_{2} + 4k_{3} + 3k_{4} + 2k_{5} + k_{6} } \right)}} \\ \end{aligned} $$

$$ \begin{aligned} G_{4n + 1}^{\left( 0 \right)} \left( \zeta \right) & : = \mathop {P.P.}\limits_{\zeta = \,\,0} \left[ {\left( {\frac{1}{\zeta } + c_{1} \zeta + c_{2} \zeta^{3} + c_{3} \zeta^{5} + c_{4} \zeta^{7} + c_{5} \zeta^{9} + c_{6} \zeta^{11} } \right)^{4n + 1} } \right]{\kern 1pt} \\ & = \sum\limits_{{k_{1} = 0}}^{4n + 1} { \cdot \sum\limits_{{k_{2} = 0}}^{{4n + 1 - k_{1} }} { \cdot \sum\limits_{{k_{3} = 0}}^{{4n + 1 - \,\,\left( {k_{1} + k_{2} } \right)}} { \cdot \sum\limits_{{k_{4} = 0}}^{{4n + 1 - \,\,\left( {k_{1} + k_{2} + k_{3} } \right)}} { \cdot \sum\limits_{{k_{5} = 0}}^{{4n + 1 - \,\,\left( {k_{1} + k_{2} + k_{3} + k_{4} } \right)}} { \cdot \sum\limits_{{k_{6} = 0}}^{{2n\, - \,\,\left( {6k_{1} + 5k_{2} + 4k_{3} + 3k_{4} + 2k_{5} } \right)}} {} } } } } } \\ & \quad \frac{{\left( {4n + 1} \right)!\,\,c_{6}^{{k_{1} }} c_{5}^{{k_{2} }} c_{4}^{{k_{3} }} c_{3}^{{k_{4} }} c_{2}^{{k_{5} }} c_{1}^{{k_{6} }} }}{{k_{1} !k_{2} !k_{3} !k_{4} !k_{5} !k_{6} !\left( {4n + 1 - k_{1} - k_{2} - k_{3} - k_{4} - k_{5} - k_{6} } \right)!\,}}\zeta^{{\, - \,\left( {4n + 1} \right)\, + 2\,\left( {6k_{1} + 5k_{2} + 4k_{3} + 3k_{4} + 2k_{5} + k_{6} } \right)}} \\ \end{aligned} $$

$$ \begin{aligned} G_{4n + 3}^{\left( \infty \right)} \left( \zeta \right) & : = \mathop {P.P.}\limits_{\zeta = \,\,\infty } \left[ {\left( {\zeta + \frac{{c_{1} }}{\zeta } + \frac{{c_{2} }}{{\zeta^{3} }} + \frac{{c_{3} }}{{\zeta^{5} }} + \frac{{c_{4} }}{{\zeta^{7} }} + \frac{{c_{5} }}{{\zeta^{9} }} + \frac{{c_{6} }}{{\zeta^{11} }}} \right)^{4n + 3} } \right]{\kern 1pt} \\ & = \sum\limits_{{k_{1} = 0}}^{4n + 3} { \cdot \sum\limits_{{k_{2} = 0}}^{{4n + 3 - k_{1} }} { \cdot \sum\limits_{{k_{3} = 0}}^{{4n + 3\, - \,\,\left( {k_{1} + k_{2} } \right)}} { \cdot \sum\limits_{{k_{4} = 0}}^{{4n + 3\, - \,\,\left( {k_{1} + k_{2} + k_{3} } \right)}} { \cdot \sum\limits_{{k_{5} = 0}}^{{4n + 3\, - \,\,\left( {k_{1} + k_{2} + k_{3} + k_{4} } \right)}} { \cdot \sum\limits_{{k_{6} = 0}}^{{\frac{4n + 3}{2}\,\, - \,\,\left( {6k_{1} + 5k_{2} + 4k_{3} + 3k_{4} + 2k_{5} } \right)}} {} } } } } } \\ & \quad \frac{{\left( {4n + 3} \right)!\,\,c_{6}^{{k_{1} }} c_{5}^{{k_{2} }} c_{4}^{{k_{3} }} c_{3}^{{k_{4} }} c_{2}^{{k_{5} }} c_{1}^{{k_{6} }} }}{{k_{1} !k_{2} !k_{3} !k_{4} !k_{5} !k_{6} !\left( {4n + 3 - k_{1} - k_{2} - k_{3} - k_{4} - k_{5} - k_{6} } \right)!\,}}\zeta^{{\,4n + 3\, - \,2\,\left( {6k_{1} + 5k_{2} + 4k_{3} + 3k_{4} + 2k_{5} + k_{6} } \right)}} \\ \end{aligned} $$

$$ \begin{aligned} G_{{2\left( {2n + 1} \right)}}^{\left( 0 \right)} \left( \zeta \right): = & \mathop {P.P.}\limits_{\zeta = \,\,0} \left[ {\left( {\zeta + \frac{{c_{1} }}{\zeta } + \frac{{c_{2} }}{{\zeta^{3} }} + \frac{{c_{3} }}{{\zeta^{5} }} + \frac{{c_{4} }}{{\zeta^{7} }} + \frac{{c_{5} }}{{\zeta^{9} }} + \frac{{c_{6} }}{{\zeta^{11} }}} \right)\left( {\frac{1}{\zeta } + c_{1} \zeta + c_{2} \zeta^{3} + c_{3} \zeta^{5} + c_{4} \zeta^{7} + c_{5} \zeta^{9} + c_{6} \zeta^{11} } \right)^{{2\left( {2n + 1} \right)}} } \right]{\kern 1pt} = \\ & \quad \sum\limits_{{k_{1} = 0}}^{{2\left( {2n + 1} \right)}} { \cdot \sum\limits_{{k_{2} = 0}}^{{2\left( {2n + 1} \right)\, - \,k_{1} }} { \cdot \sum\limits_{{k_{3} = 0}}^{{2\left( {2n + 1} \right)\, - \,\,\left( {k_{1} + k_{2} } \right)}} { \cdot \sum\limits_{{k_{4} = 0}}^{{2\left( {2n + 1} \right)\, - \,\,\left( {k_{1} + k_{2} + k_{3} } \right)}} { \cdot \sum\limits_{{k_{5} = 0}}^{{2\left( {2n + 1} \right)\, - \,\,\left( {k_{1} + k_{2} + k_{3} + k_{4} } \right)}} { \cdot \sum\limits_{{k_{6} = 0}}^{{2n - \,\,\left( {6k_{1} + 5k_{2} + 4k_{3} + 3k_{4} + 2k_{5} } \right)}} {} } } } } } \\ & \quad \frac{{\left[ {2\left( {2n + 1} \right)} \right]!\,\,c_{6}^{{k_{1} }} c_{5}^{{k_{2} }} c_{4}^{{k_{3} }} c_{3}^{{k_{4} }} c_{2}^{{k_{5} }} c_{1}^{{k_{6} }} }}{{k_{1} !k_{2} !k_{3} !k_{4} !k_{5} !k_{6} !\left[ {2\left( {2n + 1} \right) - k_{1} - k_{2} - k_{3} - k_{4} - k_{5} - k_{6} } \right]!\,}}\zeta^{{ - \,\,\left( {4n + 1} \right)\, + 2\,\left( {6k_{1} + 5k_{2} + 4k_{3} + 3k_{4} + 2k_{5} + k_{6} } \right)}} + \\ & \quad \sum\limits_{{k_{1} = 0}}^{{2\left( {2n + 1} \right)}} { \cdot \sum\limits_{{k_{2} = 0}}^{{2\left( {2n + 1} \right)\, - \,k_{1} }} { \cdot \sum\limits_{{k_{3} = 0}}^{{2\left( {2n + 1} \right)\, - \,\,\left( {k_{1} + k_{2} } \right)}} { \cdot \sum\limits_{{k_{4} = 0}}^{{2\left( {2n + 1} \right)\, - \,\,\left( {k_{1} + k_{2} + k_{3} } \right)}} { \cdot \sum\limits_{{k_{5} = 0}}^{{2\left( {2n + 1} \right)\, - \,\,\left( {k_{1} + k_{2} + k_{3} + k_{4} } \right)}} { \cdot \sum\limits_{{k_{6} = 0}}^{{2n + 1\, - \,\,\left( {6k_{1} + 5k_{2} + 4k_{3} + 3k_{4} + 2k_{5} } \right)}} {} } } } } } \\ & \quad \frac{{\left[ {2\left( {2n + 1} \right)} \right]!\,\,c_{6}^{{k_{1} }} c_{5}^{{k_{2} }} c_{4}^{{k_{3} }} c_{3}^{{k_{4} }} c_{2}^{{k_{5} }} c_{1}^{{k_{6} + 1}} }}{{k_{1} !k_{2} !k_{3} !k_{4} !k_{5} !k_{6} !\left[ {2\left( {2n + 1} \right) - k_{1} - k_{2} - k_{3} - k_{4} - k_{5} - k_{6} } \right]!\,}}\zeta^{{\, - \,\,\left( {4n + 3} \right) + \,2\,\left( {6k_{1} + 5k_{2} + 4k_{3} + 3k_{4} + 2k_{5} + k_{6} } \right)}} \\ & + \sum\limits_{{k_{1} = 0}}^{{2\left( {2n + 1} \right)}} { \cdot \sum\limits_{{k_{2} = 0}}^{{2\left( {2n + 1} \right)\, - \,k_{1} }} { \cdot \sum\limits_{{k_{3} = 0}}^{{2\left( {2n + 1} \right)\, - \,\,\left( {k_{1} + k_{2} } \right)}} { \cdot \sum\limits_{{k_{4} = 0}}^{{2\left( {2n + 1} \right)\, - \,\,\left( {k_{1} + k_{2} + k_{3} } \right)}} { \cdot \sum\limits_{{k_{5} = 0}}^{{2\left( {2n + 1} \right)\, - \,\,\left( {k_{1} + k_{2} + k_{3} + k_{4} } \right)}} { \cdot \sum\limits_{{k_{6} = 0}}^{{2\,\left( {n + 1} \right) - \,\,\left( {6k_{1} + 5k_{2} + 4k_{3} + 3k_{4} + 2k_{5} } \right)}} {} } } } } } \\ & \quad \frac{{\left[ {2\left( {2n + 1} \right)} \right]!\,\,c_{6}^{{k_{1} }} c_{5}^{{k_{2} }} c_{4}^{{k_{3} }} c_{3}^{{k_{4} }} c_{2}^{{k_{5} + 1}} c_{1}^{{k_{6} }} }}{{k_{1} !k_{2} !k_{3} !k_{4} !k_{5} !k_{6} !\left[ {2\left( {2n + 1} \right) - k_{1} - k_{2} - k_{3} - k_{4} - k_{5} - k_{6} } \right]!\,}}\zeta^{{\, - \,\,\left( {4n + 5} \right) + \,2\,\left( {6k_{1} + 5k_{2} + 4k_{3} + 3k_{4} + 2k_{5} + k_{6} } \right)}} + \\ & \quad \sum\limits_{{k_{1} = 0}}^{{2\left( {2n + 1} \right)}} { \cdot \sum\limits_{{k_{2} = 0}}^{{2\left( {2n + 1} \right)\, - \,k_{1} }} { \cdot \sum\limits_{{k_{3} = 0}}^{{2\left( {2n + 1} \right)\, - \,\,\left( {k_{1} + k_{2} } \right)}} { \cdot \sum\limits_{{k_{4} = 0}}^{{2\left( {2n + 1} \right)\, - \,\,\left( {k_{1} + k_{2} + k_{3} } \right)}} { \cdot \sum\limits_{{k_{5} = 0}}^{{2\left( {2n + 1} \right)\, - \,\,\left( {k_{1} + k_{2} + k_{3} + k_{4} } \right)}} { \cdot \sum\limits_{{k_{6} = 0}}^{{2\,n + 3\, - \,\,\left( {6k_{1} + 5k_{2} + 4k_{3} + 3k_{4} + 2k_{5} } \right)}} {} } } } } } \\ & \quad \frac{{\left[ {2\left( {2n + 1} \right)} \right]!\,\,c_{6}^{{k_{1} }} c_{5}^{{k_{2} }} c_{4}^{{k_{3} }} c_{3}^{{k_{4} + 1}} c_{2}^{{k_{5} }} c_{1}^{{k_{6} }} }}{{k_{1} !k_{2} !k_{3} !k_{4} !k_{5} !k_{6} !\left[ {2\left( {2n + 1} \right) - k_{1} - k_{2} - k_{3} - k_{4} - k_{5} - k_{6} } \right]!\,}}\zeta^{{\, - \,\,\left( {4n + 7} \right) + \,2\,\left( {6k_{1} + 5k_{2} + 4k_{3} + 3k_{4} + 2k_{5} + k_{6} } \right)}} + \\ & \quad \sum\limits_{{k_{1} = 0}}^{{2\left( {2n + 1} \right)}} { \cdot \sum\limits_{{k_{2} = 0}}^{{2\left( {2n + 1} \right)\, - \,k_{1} }} { \cdot \sum\limits_{{k_{3} = 0}}^{{2\left( {2n + 1} \right)\, - \,\,\left( {k_{1} + k_{2} } \right)}} { \cdot \sum\limits_{{k_{4} = 0}}^{{2\left( {2n + 1} \right)\, - \,\,\left( {k_{1} + k_{2} + k_{3} } \right)}} { \cdot \sum\limits_{{k_{5} = 0}}^{{2\left( {2n + 1} \right)\, - \,\,\left( {k_{1} + k_{2} + k_{3} + k_{4} } \right)}} { \cdot \sum\limits_{{k_{6} = 0}}^{{2\,\,\left( {n + 2} \right)\, - \,\,\left( {6k_{1} + 5k_{2} + 4k_{3} + 3k_{4} + 2k_{5} } \right)}} {} } } } } } \\ & \quad \frac{{\left[ {2\left( {2n + 1} \right)} \right]!\,\,c_{6}^{{k_{1} }} c_{5}^{{k_{2} }} c_{4}^{{k_{3} + 1}} c_{3}^{{k_{4} }} c_{2}^{{k_{5} }} c_{1}^{{k_{6} }} }}{{k_{1} !k_{2} !k_{3} !k_{4} !k_{5} !k_{6} !\left[ {2\left( {2n + 1} \right) - k_{1} - k_{2} - k_{3} - k_{4} - k_{5} - k_{6} } \right]!\,}}\zeta^{{\, - \,\,\left( {4n + 9} \right) + \,2\,\left( {6k_{1} + 5k_{2} + 4k_{3} + 3k_{4} + 2k_{5} + k_{6} } \right)}} + \\ & \quad \sum\limits_{{k_{1} = 0}}^{{2\left( {2n + 1} \right)}} { \cdot \sum\limits_{{k_{2} = 0}}^{{2\left( {2n + 1} \right)\, - \,k_{1} }} { \cdot \sum\limits_{{k_{3} = 0}}^{{2\left( {2n + 1} \right)\, - \,\,\left( {k_{1} + k_{2} } \right)}} { \cdot \sum\limits_{{k_{4} = 0}}^{{2\left( {2n + 1} \right)\, - \,\,\left( {k_{1} + k_{2} + k_{3} } \right)}} { \cdot \sum\limits_{{k_{5} = 0}}^{{2\left( {2n + 1} \right)\, - \,\,\left( {k_{1} + k_{2} + k_{3} + k_{4} } \right)}} { \cdot \sum\limits_{{k_{6} = 0}}^{{2n + 5\, - \,\,\left( {6k_{1} + 5k_{2} + 4k_{3} + 3k_{4} + 2k_{5} } \right)}} {} } } } } } \\ & \quad \frac{{\left[ {2\left( {2n + 1} \right)} \right]!\,\,c_{6}^{{k_{1} }} c_{5}^{{k_{2} + 1}} c_{4}^{{k_{3} }} c_{3}^{{k_{4} }} c_{2}^{{k_{5} }} c_{1}^{{k_{6} }} }}{{k_{1} !k_{2} !k_{3} !k_{4} !k_{5} !k_{6} !\left[ {2\left( {2n + 1} \right) - k_{1} - k_{2} - k_{3} - k_{4} - k_{5} - k_{6} } \right]!\,}}\zeta^{{\, - \,\,\left( {4n + 11} \right) + \,2\,\left( {6k_{1} + 5k_{2} + 4k_{3} + 3k_{4} + 2k_{5} + k_{6} } \right)}} + \\ & \quad \sum\limits_{{k_{1} = 0}}^{{2\left( {2n + 1} \right)}} { \cdot \sum\limits_{{k_{2} = 0}}^{{2\left( {2n + 1} \right)\, - \,k_{1} }} { \cdot \sum\limits_{{k_{3} = 0}}^{{2\left( {2n + 1} \right)\, - \,\,\left( {k_{1} + k_{2} } \right)}} { \cdot \sum\limits_{{k_{4} = 0}}^{{2\left( {2n + 1} \right)\, - \,\,\left( {k_{1} + k_{2} + k_{3} } \right)}} { \cdot \sum\limits_{{k_{5} = 0}}^{{2\left( {2n + 1} \right)\, - \,\,\left( {k_{1} + k_{2} + k_{3} + k_{4} } \right)}} { \cdot \sum\limits_{{k_{6} = 0}}^{{2\,\,\left( {n + 3\,} \right) - \,\,\left( {6k_{1} + 5k_{2} + 4k_{3} + 3k_{4} + 2k_{5} } \right)}} {} } } } } } \\ & \quad \frac{{\left[ {2\left( {2n + 1} \right)} \right]!\,\,c_{6}^{{k_{1} + 1}} c_{5}^{{k_{2} }} c_{4}^{{k_{3} }} c_{3}^{{k_{4} }} c_{2}^{{k_{5} }} c_{1}^{{k_{6} }} }}{{k_{1} !k_{2} !k_{3} !k_{4} !k_{5} !k_{6} !\left[ {2\left( {2n + 1} \right) - k_{1} - k_{2} - k_{3} - k_{4} - k_{5} - k_{6} } \right]!\,}}\zeta^{{\, - \,\,\left( {4n + 13} \right) + \,2\,\left( {6k_{1} + 5k_{2} + 4k_{3} + 3k_{4} + 2k_{5} + k_{6} } \right)}} \\ \end{aligned} $$

$$ \begin{aligned} G_{4n + 3}^{\left( 0 \right)} \left( \zeta \right) & : = \mathop {P.P.}\limits_{\zeta = \,\,0} \left[ {\left( {\frac{1}{\zeta } + c_{1} \zeta + c_{2} \zeta^{3} + c_{3} \zeta^{5} + c_{4} \zeta^{7} + c_{5} \zeta^{9} + c_{6} \zeta^{11} } \right)^{4n + 3} } \right]{\kern 1pt} \\ & = \sum\limits_{{k_{1} = 0}}^{4n + 3} { \cdot \sum\limits_{{k_{2} = 0}}^{{4n + 3\, - \,k_{1} }} { \cdot \sum\limits_{{k_{3} = 0}}^{{4n + 3\, - \,\,\left( {k_{1} + k_{2} } \right)}} { \cdot \sum\limits_{{k_{4} = 0}}^{{4n + 3\, - \,\,\left( {k_{1} + k_{2} + k_{3} } \right)}} { \cdot \sum\limits_{{k_{5} = 0}}^{{4n + 3\, - \,\,\left( {k_{1} + k_{2} + k_{3} + k_{4} } \right)}} { \cdot \sum\limits_{{k_{6} = 0}}^{{2n\, + 1\, - \,\,\left( {6k_{1} + 5k_{2} + 4k_{3} + 3k_{4} + 2k_{5} } \right)}} {} } } } } } \\ & \quad \frac{{\left( {4n + 3} \right)!\,\,c_{6}^{{k_{1} }} c_{5}^{{k_{2} }} c_{4}^{{k_{3} }} c_{3}^{{k_{4} }} c_{2}^{{k_{5} }} c_{1}^{{k_{6} }} }}{{k_{1} !k_{2} !k_{3} !k_{4} !k_{5} !k_{6} !\left( {4n + 3 - k_{1} - k_{2} - k_{3} - k_{4} - k_{5} - k_{6} } \right)!\,}}\zeta^{{\, - \,\left( {4n + 3} \right)\, + 2\,\left( {6k_{1} + 5k_{2} + 4k_{3} + 3k_{4} + 2k_{5} + k_{6} } \right)}} \\ \end{aligned} $$

$$ \begin{aligned} G_{4n}^{\left( \infty \right)} \left( \zeta \right): = & \mathop {P.P.}\limits_{\zeta = \,\,\infty } \left[ {\left( {\frac{1}{\zeta } + c_{1} \zeta + c_{2} \zeta^{3} + c_{3} \zeta^{5} + c_{4} \zeta^{7} + c_{5} \zeta^{9} + c_{6} \zeta^{11} } \right)\left( {\zeta + \frac{{c_{1} }}{\zeta } + \frac{{c_{2} }}{{\zeta^{3} }} + \frac{{c_{3} }}{{\zeta^{5} }} + \frac{{c_{4} }}{{\zeta^{7} }} + \frac{{c_{5} }}{{\zeta^{9} }} + \frac{{c_{6} }}{{\zeta^{11} }}} \right)^{4n} } \right]{\kern 1pt} = \\ & \quad \sum\limits_{{k_{1} = 0}}^{4n} { \cdot \sum\limits_{{k_{2} = 0}}^{{4n - \,k_{1} }} { \cdot \sum\limits_{{k_{3} = 0}}^{{4n - \,\,\left( {k_{1} + k_{2} } \right)}} { \cdot \sum\limits_{{k_{4} = 0}}^{{4n - \,\,\left( {k_{1} + k_{2} + k_{3} } \right)}} { \cdot \sum\limits_{{k_{5} = 0}}^{{4n - \,\,\left( {k_{1} + k_{2} + k_{3} + k_{4} } \right)}} { \cdot \sum\limits_{{k_{6} = 0}}^{{\frac{4n - \,1}{2}\, - \,\,\left( {6k_{1} + 5k_{2} + 4k_{3} + 3k_{4} + 2k_{5} } \right)}} {} } } } } } \\ & \quad \frac{{\left( {4n} \right)!\,\,c_{6}^{{k_{1} }} c_{5}^{{k_{2} }} c_{4}^{{k_{3} }} c_{3}^{{k_{4} }} c_{2}^{{k_{5} }} c_{1}^{{k_{6} }} }}{{k_{1} !k_{2} !k_{3} !k_{4} !k_{5} !k_{6} !\left( {4n - k_{1} - k_{2} - k_{3} - k_{4} - k_{5} - k_{6} } \right)!\,}}\zeta^{{4n - \,1\, - 2\,\left( {6k_{1} + 5k_{2} + 4k_{3} + 3k_{4} + 2k_{5} + k_{6} } \right)}} + \\ & + \sum\limits_{{k_{1} = 0}}^{4n} { \cdot \sum\limits_{{k_{2} = 0}}^{{4n - \,k_{1} }} { \cdot \sum\limits_{{k_{3} = 0}}^{{4n - \,\,\left( {k_{1} + k_{2} } \right)}} { \cdot \sum\limits_{{k_{4} = 0}}^{{4n - \,\,\left( {k_{1} + k_{2} + k_{3} } \right)}} { \cdot \sum\limits_{{k_{5} = 0}}^{{4n - \,\,\left( {k_{1} + k_{2} + k_{3} + k_{4} } \right)}} { \cdot \sum\limits_{{k_{6} = 0}}^{{\frac{4n + \,1}{2}\, - \,\,\left( {6k_{1} + 5k_{2} + 4k_{3} + 3k_{4} + 2k_{5} } \right)}} {} } } } } } \\ & \quad \frac{{\left( {4n} \right)!\,\,c_{6}^{{k_{1} }} c_{5}^{{k_{2} }} c_{4}^{{k_{3} }} c_{3}^{{k_{4} }} c_{2}^{{k_{5} }} c_{1}^{{k_{6} + 1}} }}{{k_{1} !k_{2} !k_{3} !k_{4} !k_{5} !k_{6} !\left( {4n - k_{1} - k_{2} - k_{3} - k_{4} - k_{5} - k_{6} } \right)!\,}}\zeta^{{4n + \,1\, - 2\,\left( {6k_{1} + 5k_{2} + 4k_{3} + 3k_{4} + 2k_{5} + k_{6} } \right)}} + \\ & \quad \sum\limits_{{k_{1} = 0}}^{4n} { \cdot \sum\limits_{{k_{2} = 0}}^{{4n - \,k_{1} }} { \cdot \sum\limits_{{k_{3} = 0}}^{{4n - \,\,\left( {k_{1} + k_{2} } \right)}} { \cdot \sum\limits_{{k_{4} = 0}}^{{4n - \,\,\left( {k_{1} + k_{2} + k_{3} } \right)}} { \cdot \sum\limits_{{k_{5} = 0}}^{{4n - \,\,\left( {k_{1} + k_{2} + k_{3} + k_{4} } \right)}} { \cdot \sum\limits_{{k_{6} = 0}}^{{\frac{4n + \,3}{2}\, - \,\,\left( {6k_{1} + 5k_{2} + 4k_{3} + 3k_{4} + 2k_{5} } \right)}} {} } } } } } \\ & \quad \frac{{\left( {4n} \right)!\,\,c_{6}^{{k_{1} }} c_{5}^{{k_{2} }} c_{4}^{{k_{3} }} c_{3}^{{k_{4} }} c_{2}^{{k_{5} + 1}} c_{1}^{{k_{6} }} }}{{k_{1} !k_{2} !k_{3} !k_{4} !k_{5} !k_{6} !\left( {4n - k_{1} - k_{2} - k_{3} - k_{4} - k_{5} - k_{6} } \right)!\,}}\zeta^{{4n + \,3\, - 2\,\left( {6k_{1} + 5k_{2} + 4k_{3} + 3k_{4} + 2k_{5} + k_{6} } \right)}} + \\ & \quad \sum\limits_{{k_{1} = 0}}^{4n} { \cdot \sum\limits_{{k_{2} = 0}}^{{4n - \,k_{1} }} { \cdot \sum\limits_{{k_{3} = 0}}^{{4n - \,\,\left( {k_{1} + k_{2} } \right)}} { \cdot \sum\limits_{{k_{4} = 0}}^{{4n - \,\,\left( {k_{1} + k_{2} + k_{3} } \right)}} { \cdot \sum\limits_{{k_{5} = 0}}^{{4n - \,\,\left( {k_{1} + k_{2} + k_{3} + k_{4} } \right)}} { \cdot \sum\limits_{{k_{6} = 0}}^{{\frac{4n + \,5}{2}\, - \,\,\left( {6k_{1} + 5k_{2} + 4k_{3} + 3k_{4} + 2k_{5} } \right)}} {} } } } } } \\ & \quad \frac{{\left( {4n} \right)!\,\,c_{6}^{{k_{1} }} c_{5}^{{k_{2} }} c_{4}^{{k_{3} }} c_{3}^{{k_{4} + 1}} c_{2}^{{k_{5} }} c_{1}^{{k_{6} }} }}{{k_{1} !k_{2} !k_{3} !k_{4} !k_{5} !k_{6} !\left( {4n - k_{1} - k_{2} - k_{3} - k_{4} - k_{5} - k_{6} } \right)!\,}}\zeta^{{4n + \,5\, - 2\,\left( {6k_{1} + 5k_{2} + 4k_{3} + 3k_{4} + 2k_{5} + k_{6} } \right)}} + \\ & \quad \sum\limits_{{k_{1} = 0}}^{4n} { \cdot \sum\limits_{{k_{2} = 0}}^{{4n - \,k_{1} }} { \cdot \sum\limits_{{k_{3} = 0}}^{{4n - \,\,\left( {k_{1} + k_{2} } \right)}} { \cdot \sum\limits_{{k_{4} = 0}}^{{4n - \,\,\left( {k_{1} + k_{2} + k_{3} } \right)}} { \cdot \sum\limits_{{k_{5} = 0}}^{{4n - \,\,\left( {k_{1} + k_{2} + k_{3} + k_{4} } \right)}} { \cdot \sum\limits_{{k_{6} = 0}}^{{\frac{4n + \,7}{2}\, - \,\,\left( {6k_{1} + 5k_{2} + 4k_{3} + 3k_{4} + 2k_{5} } \right)}} {} } } } } } \\ & \quad \frac{{\left( {4n} \right)!\,\,c_{6}^{{k_{1} }} c_{5}^{{k_{2} }} c_{4}^{{k_{3} + 1}} c_{3}^{{k_{4} }} c_{2}^{{k_{5} }} c_{1}^{{k_{6} }} }}{{k_{1} !k_{2} !k_{3} !k_{4} !k_{5} !k_{6} !\left( {4n - k_{1} - k_{2} - k_{3} - k_{4} - k_{5} - k_{6} } \right)!\,}}\zeta^{{4n + \,7\, - 2\,\left( {6k_{1} + 5k_{2} + 4k_{3} + 3k_{4} + 2k_{5} + k_{6} } \right)}} + \\ & \quad \sum\limits_{{k_{1} = 0}}^{4n} { \cdot \sum\limits_{{k_{2} = 0}}^{{4n - \,k_{1} }} { \cdot \sum\limits_{{k_{3} = 0}}^{{4n - \,\,\left( {k_{1} + k_{2} } \right)}} { \cdot \sum\limits_{{k_{4} = 0}}^{{4n - \,\,\left( {k_{1} + k_{2} + k_{3} } \right)}} { \cdot \sum\limits_{{k_{5} = 0}}^{{4n - \,\,\left( {k_{1} + k_{2} + k_{3} + k_{4} } \right)}} { \cdot \sum\limits_{{k_{6} = 0}}^{{\frac{4n + \,9}{2}\, - \,\,\left( {6k_{1} + 5k_{2} + 4k_{3} + 3k_{4} + 2k_{5} } \right)}} {} } } } } } \\ & \quad \frac{{\left( {4n} \right)!\,\,c_{6}^{{k_{1} }} c_{5}^{{k_{2} + 1}} c_{4}^{{k_{3} }} c_{3}^{{k_{4} }} c_{2}^{{k_{5} }} c_{1}^{{k_{6} }} }}{{k_{1} !k_{2} !k_{3} !k_{4} !k_{5} !k_{6} !\left( {4n - k_{1} - k_{2} - k_{3} - k_{4} - k_{5} - k_{6} } \right)!\,}}\zeta^{{4n + \,9\, - 2\,\left( {6k_{1} + 5k_{2} + 4k_{3} + 3k_{4} + 2k_{5} + k_{6} } \right)}} + \\ & \quad \sum\limits_{{k_{1} = 0}}^{4n} { \cdot \sum\limits_{{k_{2} = 0}}^{{4n - \,k_{1} }} { \cdot \sum\limits_{{k_{3} = 0}}^{{4n - \,\,\left( {k_{1} + k_{2} } \right)}} { \cdot \sum\limits_{{k_{4} = 0}}^{{4n - \,\,\left( {k_{1} + k_{2} + k_{3} } \right)}} { \cdot \sum\limits_{{k_{5} = 0}}^{{4n - \,\,\left( {k_{1} + k_{2} + k_{3} + k_{4} } \right)}} { \cdot \sum\limits_{{k_{6} = 0}}^{{\frac{4n + \,11}{2}\, - \,\,\left( {6k_{1} + 5k_{2} + 4k_{3} + 3k_{4} + 2k_{5} } \right)}} {} } } } } } \\ & \quad \frac{{\left( {4n} \right)!\,\,c_{6}^{{k_{1} + 1}} c_{5}^{{k_{2} }} c_{4}^{{k_{3} }} c_{3}^{{k_{4} }} c_{2}^{{k_{5} }} c_{1}^{{k_{6} }} }}{{k_{1} !k_{2} !k_{3} !k_{4} !k_{5} !k_{6} !\left( {4n - k_{1} - k_{2} - k_{3} - k_{4} - k_{5} - k_{6} } \right)!\,}}\zeta^{{4n + \,11\, - 2\,\left( {6k_{1} + 5k_{2} + 4k_{3} + 3k_{4} + 2k_{5} + k_{6} } \right)}} \\ \end{aligned} $$

$$ \begin{aligned} G_{{2\left( {2n + 1} \right)}}^{\left( \infty \right)} \left( \zeta \right): = & \mathop {P.P.}\limits_{\zeta = \,\,\infty } \left[ {\left( {\frac{1}{\zeta } + c_{1} \zeta + c_{2} \zeta^{3} + c_{3} \zeta^{5} + c_{4} \zeta^{7} + c_{5} \zeta^{9} + c_{6} \zeta^{11} } \right)\left( {\zeta + \frac{{c_{1} }}{\zeta } + \frac{{c_{2} }}{{\zeta^{3} }} + \frac{{c_{3} }}{{\zeta^{5} }} + \frac{{c_{4} }}{{\zeta^{7} }} + \frac{{c_{5} }}{{\zeta^{9} }} + \frac{{c_{6} }}{{\zeta^{11} }}} \right)^{{2\left( {2n + 1} \right)}} } \right]{\kern 1pt} = \\ & \quad \sum\limits_{{k_{1} = 0}}^{{2\left( {2n + 1} \right)}} { \cdot \sum\limits_{{k_{2} = 0}}^{{2\left( {2n + 1} \right)\, - \,k_{1} }} { \cdot \sum\limits_{{k_{3} = 0}}^{{2\left( {2n + 1} \right)\, - \,\,\left( {k_{1} + k_{2} } \right)}} { \cdot \sum\limits_{{k_{4} = 0}}^{{2\left( {2n + 1} \right)\, - \,\,\left( {k_{1} + k_{2} + k_{3} } \right)}} { \cdot \sum\limits_{{k_{5} = 0}}^{{2\left( {2n + 1} \right)\, - \,\,\left( {k_{1} + k_{2} + k_{3} + k_{4} } \right)}} { \cdot \sum\limits_{{k_{6} = 0}}^{{\frac{4n + 1}{2}\, - \,\,\left( {6k_{1} + 5k_{2} + 4k_{3} + 3k_{4} + 2k_{5} } \right)}} {} } } } } } \\ & \quad \frac{{\left[ {2\left( {2n + 1} \right)} \right]!\,\,c_{6}^{{k_{1} }} c_{5}^{{k_{2} }} c_{4}^{{k_{3} }} c_{3}^{{k_{4} }} c_{2}^{{k_{5} }} c_{1}^{{k_{6} }} }}{{k_{1} !k_{2} !k_{3} !k_{4} !k_{5} !k_{6} !\left[ {2\left( {2n + 1} \right) - k_{1} - k_{2} - k_{3} - k_{4} - k_{5} - k_{6} } \right]!\,}}\zeta^{{4n + 1\, - 2\,\left( {6k_{1} + 5k_{2} + 4k_{3} + 3k_{4} + 2k_{5} + k_{6} } \right)}} + \\ & + \sum\limits_{{k_{1} = 0}}^{{2\left( {2n + 1} \right)}} { \cdot \sum\limits_{{k_{2} = 0}}^{{2\left( {2n + 1} \right)\, - \,k_{1} }} { \cdot \sum\limits_{{k_{3} = 0}}^{{2\left( {2n + 1} \right)\, - \,\,\left( {k_{1} + k_{2} } \right)}} { \cdot \sum\limits_{{k_{4} = 0}}^{{2\left( {2n + 1} \right)\, - \,\,\left( {k_{1} + k_{2} + k_{3} } \right)}} { \cdot \sum\limits_{{k_{5} = 0}}^{{2\left( {2n + 1} \right)\, - \,\,\left( {k_{1} + k_{2} + k_{3} + k_{4} } \right)}} { \cdot \sum\limits_{{k_{6} = 0}}^{{\frac{4n + 3}{2}\, - \,\,\left( {6k_{1} + 5k_{2} + 4k_{3} + 3k_{4} + 2k_{5} } \right)}} {} } } } } } \\ & \quad \frac{{\left[ {2\left( {2n + 1} \right)} \right]!\,\,c_{6}^{{k_{1} }} c_{5}^{{k_{2} }} c_{4}^{{k_{3} }} c_{3}^{{k_{4} }} c_{2}^{{k_{5} }} c_{1}^{{k_{6} + 1}} }}{{k_{1} !k_{2} !k_{3} !k_{4} !k_{5} !k_{6} !\left[ {2\left( {2n + 1} \right) - k_{1} - k_{2} - k_{3} - k_{4} - k_{5} - k_{6} } \right]!\,}}\zeta^{{4n + 3\, - 2\,\left( {6k_{1} + 5k_{2} + 4k_{3} + 3k_{4} + 2k_{5} + k_{6} } \right)}} + \\ & \quad \sum\limits_{{k_{1} = 0}}^{{2\left( {2n + 1} \right)}} { \cdot \sum\limits_{{k_{2} = 0}}^{{2\left( {2n + 1} \right)\, - \,k_{1} }} { \cdot \sum\limits_{{k_{3} = 0}}^{{2\left( {2n + 1} \right)\, - \,\,\left( {k_{1} + k_{2} } \right)}} { \cdot \sum\limits_{{k_{4} = 0}}^{{2\left( {2n + 1} \right)\, - \,\,\left( {k_{1} + k_{2} + k_{3} } \right)}} { \cdot \sum\limits_{{k_{5} = 0}}^{{2\left( {2n + 1} \right)\, - \,\,\left( {k_{1} + k_{2} + k_{3} + k_{4} } \right)}} { \cdot \sum\limits_{{k_{6} = 0}}^{{\frac{4n + 5}{2}\, - \,\,\left( {6k_{1} + 5k_{2} + 4k_{3} + 3k_{4} + 2k_{5} } \right)}} {} } } } } } \\ & \quad \frac{{\left[ {2\left( {2n + 1} \right)} \right]!\,\,c_{6}^{{k_{1} }} c_{5}^{{k_{2} }} c_{4}^{{k_{3} }} c_{3}^{{k_{4} }} c_{2}^{{k_{5} + 1}} c_{1}^{{k_{6} }} }}{{k_{1} !k_{2} !k_{3} !k_{4} !k_{5} !k_{6} !\left[ {2\left( {2n + 1} \right) - k_{1} - k_{2} - k_{3} - k_{4} - k_{5} - k_{6} } \right]!\,}}\zeta^{{4n + 5\, - 2\,\left( {6k_{1} + 5k_{2} + 4k_{3} + 3k_{4} + 2k_{5} + k_{6} } \right)}} + \\ & \quad \sum\limits_{{k_{1} = 0}}^{{2\left( {2n + 1} \right)}} { \cdot \sum\limits_{{k_{2} = 0}}^{{2\left( {2n + 1} \right)\, - \,k_{1} }} { \cdot \sum\limits_{{k_{3} = 0}}^{{2\left( {2n + 1} \right)\, - \,\,\left( {k_{1} + k_{2} } \right)}} { \cdot \sum\limits_{{k_{4} = 0}}^{{2\left( {2n + 1} \right)\, - \,\,\left( {k_{1} + k_{2} + k_{3} } \right)}} { \cdot \sum\limits_{{k_{5} = 0}}^{{2\left( {2n + 1} \right)\, - \,\,\left( {k_{1} + k_{2} + k_{3} + k_{4} } \right)}} { \cdot \sum\limits_{{k_{6} = 0}}^{{\frac{4n + 7}{2}\, - \,\,\left( {6k_{1} + 5k_{2} + 4k_{3} + 3k_{4} + 2k_{5} } \right)}} {} } } } } } \\ & \quad \frac{{\left[ {2\left( {2n + 1} \right)} \right]!\,\,c_{6}^{{k_{1} }} c_{5}^{{k_{2} }} c_{4}^{{k_{3} }} c_{3}^{{k_{4} + 1}} c_{2}^{{k_{5} }} c_{1}^{{k_{6} }} }}{{k_{1} !k_{2} !k_{3} !k_{4} !k_{5} !k_{6} !\left[ {2\left( {2n + 1} \right) - k_{1} - k_{2} - k_{3} - k_{4} - k_{5} - k_{6} } \right]!\,}}\zeta^{{4n + 7\, - 2\,\left( {6k_{1} + 5k_{2} + 4k_{3} + 3k_{4} + 2k_{5} + k_{6} } \right)}} + \\ & \quad \sum\limits_{{k_{1} = 0}}^{{2\left( {2n + 1} \right)}} { \cdot \sum\limits_{{k_{2} = 0}}^{{2\left( {2n + 1} \right)\, - \,k_{1} }} { \cdot \sum\limits_{{k_{3} = 0}}^{{2\left( {2n + 1} \right)\, - \,\,\left( {k_{1} + k_{2} } \right)}} { \cdot \sum\limits_{{k_{4} = 0}}^{{2\left( {2n + 1} \right)\, - \,\,\left( {k_{1} + k_{2} + k_{3} } \right)}} { \cdot \sum\limits_{{k_{5} = 0}}^{{2\left( {2n + 1} \right)\, - \,\,\left( {k_{1} + k_{2} + k_{3} + k_{4} } \right)}} { \cdot \sum\limits_{{k_{6} = 0}}^{{\frac{4n + 9}{2}\, - \,\,\left( {6k_{1} + 5k_{2} + 4k_{3} + 3k_{4} + 2k_{5} } \right)}} {} } } } } } \\ & \quad \frac{{\left[ {2\left( {2n + 1} \right)} \right]!\,\,c_{6}^{{k_{1} }} c_{5}^{{k_{2} }} c_{4}^{{k_{3} + 1}} c_{3}^{{k_{4} }} c_{2}^{{k_{5} }} c_{1}^{{k_{6} }} }}{{k_{1} !k_{2} !k_{3} !k_{4} !k_{5} !k_{6} !\left[ {2\left( {2n + 1} \right) - k_{1} - k_{2} - k_{3} - k_{4} - k_{5} - k_{6} } \right]!\,}}\zeta^{{4n + 9\, - 2\,\left( {6k_{1} + 5k_{2} + 4k_{3} + 3k_{4} + 2k_{5} + k_{6} } \right)}} + \\ & \quad \sum\limits_{{k_{1} = 0}}^{{2\left( {2n + 1} \right)}} { \cdot \sum\limits_{{k_{2} = 0}}^{{2\left( {2n + 1} \right)\, - \,k_{1} }} { \cdot \sum\limits_{{k_{3} = 0}}^{{2\left( {2n + 1} \right)\, - \,\,\left( {k_{1} + k_{2} } \right)}} { \cdot \sum\limits_{{k_{4} = 0}}^{{2\left( {2n + 1} \right)\, - \,\,\left( {k_{1} + k_{2} + k_{3} } \right)}} { \cdot \sum\limits_{{k_{5} = 0}}^{{2\left( {2n + 1} \right)\, - \,\,\left( {k_{1} + k_{2} + k_{3} + k_{4} } \right)}} { \cdot \sum\limits_{{k_{6} = 0}}^{{\frac{4n + 11}{2}\, - \,\,\left( {6k_{1} + 5k_{2} + 4k_{3} + 3k_{4} + 2k_{5} } \right)}} {} } } } } } \\ & \quad \frac{{\left[ {2\left( {2n + 1} \right)} \right]!\,\,c_{6}^{{k_{1} }} c_{5}^{{k_{2} + 1}} c_{4}^{{k_{3} }} c_{3}^{{k_{4} }} c_{2}^{{k_{5} }} c_{1}^{{k_{6} }} }}{{k_{1} !k_{2} !k_{3} !k_{4} !k_{5} !k_{6} !\left[ {2\left( {2n + 1} \right) - k_{1} - k_{2} - k_{3} - k_{4} - k_{5} - k_{6} } \right]!\,}}\zeta^{{4n + 11\, - 2\,\left( {6k_{1} + 5k_{2} + 4k_{3} + 3k_{4} + 2k_{5} + k_{6} } \right)}} + \\ & \quad \sum\limits_{{k_{1} = 0}}^{{2\left( {2n + 1} \right)}} { \cdot \sum\limits_{{k_{2} = 0}}^{{2\left( {2n + 1} \right)\, - \,k_{1} }} { \cdot \sum\limits_{{k_{3} = 0}}^{{2\left( {2n + 1} \right)\, - \,\,\left( {k_{1} + k_{2} } \right)}} { \cdot \sum\limits_{{k_{4} = 0}}^{{2\left( {2n + 1} \right)\, - \,\,\left( {k_{1} + k_{2} + k_{3} } \right)}} { \cdot \sum\limits_{{k_{5} = 0}}^{{2\left( {2n + 1} \right)\, - \,\,\left( {k_{1} + k_{2} + k_{3} + k_{4} } \right)}} { \cdot \sum\limits_{{k_{6} = 0}}^{{\frac{4n + 13}{2}\, - \,\,\left( {6k_{1} + 5k_{2} + 4k_{3} + 3k_{4} + 2k_{5} } \right)}} {} } } } } } \\ & \quad \frac{{\left[ {2\left( {2n + 1} \right)} \right]!\,\,c_{6}^{{k_{1} + 1}} c_{5}^{{k_{2} }} c_{4}^{{k_{3} }} c_{3}^{{k_{4} }} c_{2}^{{k_{5} }} c_{1}^{{k_{6} }} }}{{k_{1} !k_{2} !k_{3} !k_{4} !k_{5} !k_{6} !\left[ {2\left( {2n + 1} \right) - k_{1} - k_{2} - k_{3} - k_{4} - k_{5} - k_{6} } \right]!\,}}\zeta^{{4n + 13\, - 2\,\left( {6k_{1} + 5k_{2} + 4k_{3} + 3k_{4} + 2k_{5} + k_{6} } \right)}} \\ \end{aligned} $$