Legendary Chi function

The legendary Chi function (after Adrien-Marie Legendre ) is a special function in mathematics .

definition

The legendary chi function is defined as follows:

{\ displaystyle \ chi _ {\ nu} (z) = \ sum _ {k = 0} ^ {\ infty} {\ frac {z ^ {2k + 1}} {(2k + 1) ^ {\ nu} }}.}

It can also be expressed with the polylogarithm : ${\ displaystyle \ mathrm {Li} _ {\ nu} (z)}$

{\ displaystyle \ chi _ {\ nu} (z) = {\ frac {1} {2}} \ left [\ operatorname {Li} _ {\ nu} (z) - \ operatorname {Li} _ {\ nu } (- z) \ right].}

{\ displaystyle {\ begin {matrix} \ chi _ {2} (\ mathrm {i}) & = & \ mathrm {i} \ cdot G \\\ chi _ {2} ({\ sqrt {2}} - 1) & = & {\ frac {1} {16}} \ pi ^ {2} - {\ frac {1} {4}} \ left (\ ln ({\ sqrt {2}} + 1) \ right ) ^ {2} \\\ chi _ {2} ({\ frac {1} {2}} ({\ sqrt {5}} - 1)) & = & {\ frac {1} {12}} \ pi ^ {2} - {\ frac {3} {4}} \ left (\ ln ({\ sqrt {5}} + 1) \ right) ^ {2} \\\ chi _ {2} ({\ sqrt {5}} - 2) & = & {\ frac {1} {24}} \ pi ^ {2} - {\ frac {3} {4}} \ left (\ ln ({\ sqrt {5} } +1) \ right) ^ {2} \\\ chi _ {2} (- 1) & = & - {\ frac {1} {8}} \ pi ^ {2} \\\ chi _ {2 } (1) & = & {\ frac {1} {8}} \ pi ^ {2} \ end {matrix}}}

The Dirichlet Lambda function is one of the special cases ${\ displaystyle \ lambda}$

{\ displaystyle \ lambda (n) = \ chi _ {n} (1) \,}

and the Dirichlet beta function : ${\ displaystyle \ beta}$

{\ displaystyle \ beta (n) = {\ frac {1} {\ rm {i}}} \ chi _ {n} (\ mathrm {i}).}

The transcendent Lerchian zeta function generalizes the legendary Chi function:

{\ displaystyle \ chi _ {n} (z) = 2 ^ {- n} z \, \ Phi (z ^ {2}, n, {\ tfrac {1} {2}}).}