# LOG#161. Polylogia flashes(III). In the third post of this series I will write more fantastic identities related to our friends, the polylogs!

(1) and by analytic continuation that equation can be extended to all . In fact

(2) such as  , since we define

(3) and the equations

(4) (5) (6) Duplication formula for the polylogarithm:

(7) Connection with the Kummer’s function can be established

(8) (9) and thus

(10) We also have

(11) Some extra values of (negative) integer polylogs that are rational functions or logarithms

(12) (13) (14) (15) (16) (17) And more generally, we have the general formulae

(18) (19) and where are the Stirling numbers of the second kind.

(20) Furthermore,

(21) and where are the eulerian numbers.

We write now some interesting values of the polylog you will love too    and where  with .

The polylog and other functions can be also be related, as we have seen:

(22) (23) (24) where is the Dirichlet beta function. The complete Fermi-Dirac integral is also polylogarithmic The incomplete polylog is also interesting   and the incomplete gamma function is defined by   (25) with , ,  . Moreover,

(26) There is also a formula called inversion formula

(27)  and for (28) The expression implies that

(29) with if and if . The following feynmanity (nullity) identity holds as well

(30)  and (31) for and

(32) for . Polylogs and Clausen functions are related (we already saw this before)

(33) The inverse tangent integral is related to polylogs

(34) (35) (36) (37) The Legendre chi function is also related to polylogs

(38) The incomplete zeta function or Debye functions are polylogs as well

(39) (40) (41) (42) This last integral converges if  and . It is the Bose-Einstein distribution!

(43) For and excepting and we have and

(44) with residue equal to See you in the next polylog post!

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