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  "body": "\n## Summary\n\nAn elliptic hypergeometric series is a series $\\sum c_n$ in which the ratio $c_{n+1}/c_n$ is an [[elliptic function|FF9ZPCZ59K]] of $n$.\nIn ordinary [[hypergeometric series|428M453CR2]] the ratio is a rational function of $n$.\nIn [[basic hypergeometric series|EB4N9RQR51]] it is a rational function of $q^n$, and the elliptic case is the next step.\nThe terms are built from theta functions in place of the factors $1 - aq^k$.\nSetting the elliptic nome $p$ to $0$ gives back basic hypergeometric series.\n\n## Explanations\n\n### L1 human\n\nMany classical series are built so that each term is the previous one times a simple factor.\nFor hypergeometric series the factor is a fraction of polynomials in the index $n$.\nFor elliptic hypergeometric series the factor is a doubly periodic function of $n$, made from theta functions.\nIn the theta-function notation, putting the extra parameter $p$ equal to $0$ turns every theta factor $\\theta(x; p)$ into $1 - x$, and the series becomes a familiar $q$-series.\nSo elliptic hypergeometric series are a common generalization of the classical and the $q$-series.\n",
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