Gauss,s Digamma Theorem
المؤلف:
Allouche, J.-P.
المصدر:
"Series and Infinite Products related to Binary Expansions of Integers." 1992. http://algo.inria.fr/seminars/sem92-93/allouche.ps.
الجزء والصفحة:
...
8-8-2019
3017
Gauss's Digamma Theorem
At rational arguments 
, the digamma function 
is given by
![psi_0(p/q)=-gamma-ln(2q)-1/2picot(p/qpi)
+2sum_(k=1)^([q/2]-1)cos((2pipk)/q)ln[sin((pik)/q)]](http://mathworld.wolfram.com/images/equations/GausssDigammaTheorem/NumberedEquation1.gif) |
(1)
|
for 
(Knuth 1997, p. 94). These give the special values
where 
is the Euler-Mascheroni constant.
REFERENCES:
Allouche, J.-P. "Series and Infinite Products related to Binary Expansions of Integers." 1992. http://algo.inria.fr/seminars/sem92-93/allouche.ps.
Böhmer, E. Differenzengleichungen und bestimmte Integrale. Leipzig, Germany: Teubner, p. 77, 1939.
Erdélyi, A.; Magnus, W.; Oberhettinger, F.; and Tricomi, F. G. "The 
Function." §1.7 in Higher Transcendental Functions, Vol. 1. New York: Krieger, pp. 15-20, 1981.
Gradshteyn, I. S. and Ryzhik, I. M. Formula 8.3636 in Tables of Integrals, Series, and Products, 6th ed. San Diego, CA: Academic Press, 2000.
Jensen, J. L. W. V. "An Elementary Exposition of the Theory of the Gamma Function." Ann. Math. 17, 124-166, 1915.
Knuth, D. E. The Art of Computer Programming, Vol. 1: Fundamental Algorithms, 3rd ed. Reading, MA: Addison-Wesley, 1997.
Kölbig, K. S. "The Polygamma Function and the Derivatives of the Cotangent Function for Rational Arguments." Report CN/96/5. CERN Computing and Networks Division, 1996.
Lösch, F. and Schoblik, F. Die Fakultät und verwandte Funktionen. Leipzig, Germany: Teubner, p. 12, 1951.
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