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Notes on A-Hypergeometric Functions [Lecture notes] by Frits Beukers

25 February 2017 adminCalculus

By Frits Beukers

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Extra info for Notes on A-Hypergeometric Functions [Lecture notes]

Sample text

Fk = xk . Symbolic Comp. 43 (2008), 377-394. Adolphson, Hypergeometric functions and rings generated by monomials. Duke Math. J. 73 (1994), 269-290. Kamp´e de F´eriet, Fonctions hyperg´eom´etriques et hypersph´eriques : polynomes d’Hermite. Paris : Gauthier-Villars, 1926. [4] P. Wolfart, Algebraic Appell-Lauricella functions. Analysis 12 (1992), 359-376. AG], to appear in Compositio Math. AC]. [7] F. Heckman, Monodromy for the hypergeometric function n Fn−1 . Invent. Math. Beukers, Gauss’ hypergeometric function, in : Arithmetic and geometry around hypergeometric functions, 23 - 42, Progr.

J. 73 (1994), 269-290. Kamp´e de F´eriet, Fonctions hyperg´eom´etriques et hypersph´eriques : polynomes d’Hermite. Paris : Gauthier-Villars, 1926. [4] P. Wolfart, Algebraic Appell-Lauricella functions. Analysis 12 (1992), 359-376. AG], to appear in Compositio Math. AC]. [7] F. Heckman, Monodromy for the hypergeometric function n Fn−1 . Invent. Math. Beukers, Gauss’ hypergeometric function, in : Arithmetic and geometry around hypergeometric functions, 23 - 42, Progr. Math 180 (2010), 589-610 . AG].

Some examples of Euler integrals Recall the classical Euler integral for 2 F1 (a, b, c|z) from Section 2. 1 would be 3-dimensional, since A ⊂ Z3 in this case. We like to show how the 3-dimensional integral can be reduced to the classical integral. The integrand of the A-hypergeometric Euler integral reads ta1 tb2 t1−c dt1 dt2 dt3 3 ∧ ∧ . 1 − t1 − t2 − t3 − zt1 t2 /t3 t1 t2 t3 Replace t2 by t2 t3 to get dt1 dt2 dt3 ta1 tb2 tb+1−c 3 ∧ ∧ . 1 − t1 − t2 t3 − t3 − zt1 t2 t1 t2 t3 Now replace t1 by t1 /(1 + zt2 ) and t3 by t3 /(1 + t2 ) to get ta1 tb+1−c (1 + zt2 )−a (1 + t2 )c−b−1 tb2 dt1 dt2 dt3 3 ∧ ∧ .

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