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A hypergeometric family over $\Q$ is a motive over the field $\Q(t)$ with coefficients in $\Q$. The families are indexed by non-overlapping ordered pairs of multisets of positive integers $(A,B)$ (called cusp-indices) with $\sum_{a \in A} \phi(a) = \sum_{b \in B} \phi(b) = d \in \Z_{\geq 1}$, where $d$ is the degree and $\phi$ is Euler's totient function.

This motive has periods which satisfy the hypergeometric differential equation $$ \left(t \prod_{i=1}^d \left(t \frac{\mathrm d}{\mathrm dt} - \alpha_i\right) - \prod_{j=1}^d \left(t \frac{\mathrm d}{\mathrm dt} + \beta_j - 1\right) \right)(y) = 0 $$ where $\alpha_1,\dots,\alpha_d$ and $\beta_1,\dots,\beta_d$ are lists obtained from $A,B$ by replacing each integer $n$ with the set of all elements of $\Q \cap (0,1]$ with denominator $n$. For $k=1,\dots,d$, one solution of this equation is the hypergeometric series $$ z^{1-\beta_k} \sum_{n=0}^\infty \frac{\prod_{i=1}^d (\alpha_i - \beta_k + 1)_n}{\prod_{j=1}^d (\beta_j - \beta_k + 1)_n} z^n $$ where $(x)_n = x(x+1)\cdots(x+n-1)$ is the Pochhammer symbol.

Knowl status:
  • Review status: beta
  • Last edited by Wanlin Li on 2019-10-04 14:18:46
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