By Frits Beukers

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**Read Online or Download Notes on A-Hypergeometric Functions [Lecture notes] PDF**

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**Additional resources for Notes on A-Hypergeometric Functions [Lecture notes]**

**Example text**

Fn be n linear forms in k variables x1 , . . , xk where k < n. For any parameters λ1 , . . , λn consider the integral ∫ I(k, n, λ) = f1λ1 · · · fnλn dx1 ∧ · · · ∧ dxk as a function of the coeﬃcients of the forms f1 , . . , fn . The integral is taken over a suitable k-cycle. After a linear change of coordinates we may assume f1 = x1 , . . , fk = xk . BEUKERS ∫ −λ n t1 k+1 · · · t−λ λk+1 n−k xλ1 1 · · · xλk k fk+1 · · · fnλn 1 − t1 − · · · − tn−k dt1 ∧ · · · ∧ dtn−k ∧ dx1 ∧ · · · ∧ dxk Integrate with respect to t1 , .

Fn be n linear forms in k variables x1 , . . , xk where k < n. For any parameters λ1 , . . , λn consider the integral ∫ I(k, n, λ) = f1λ1 · · · fnλn dx1 ∧ · · · ∧ dxk as a function of the coeﬃcients of the forms f1 , . . , fn . The integral is taken over a suitable k-cycle. After a linear change of coordinates we may assume f1 = x1 , . . , fk = xk . BEUKERS ∫ −λ n t1 k+1 · · · t−λ λk+1 n−k xλ1 1 · · · xλk k fk+1 · · · fnλn 1 − t1 − · · · − tn−k dt1 ∧ · · · ∧ dtn−k ∧ dx1 ∧ · · · ∧ dxk Integrate with respect to t1 , .

Fn be n linear forms in k variables x1 , . . , xk where k < n. For any parameters λ1 , . . , λn consider the integral ∫ I(k, n, λ) = f1λ1 · · · fnλn dx1 ∧ · · · ∧ dxk as a function of the coeﬃcients of the forms f1 , . . , fn . The integral is taken over a suitable k-cycle. After a linear change of coordinates we may assume f1 = x1 , . . , fk = xk . BEUKERS ∫ −λ n t1 k+1 · · · t−λ λk+1 n−k xλ1 1 · · · xλk k fk+1 · · · fnλn 1 − t1 − · · · − tn−k dt1 ∧ · · · ∧ dtn−k ∧ dx1 ∧ · · · ∧ dxk Integrate with respect to t1 , .