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  "body": "\n## Statement\n\nLet $a, b, c \\in \\C$ with $c \\notin \\set{1, 0, -1, -2, \\ldots}$.\nWrite $F = {}_2F_1(a, b; c; z)$ for the [[hypergeometric function|463X5V31D9]].\nWrite $F(a \\pm 1)$, $F(b \\pm 1)$, $F(c \\pm 1)$ for the six *contiguous* functions obtained by changing one parameter by $\\pm 1$.\n\n1. (Gauss) Let $G$ and $H$ be two of the six contiguous functions.\nThen $F$, $G$, $H$ satisfy a linear relation whose coefficients are polynomials in $a, b, c, z$, not all zero.\n2. In particular, for all $z \\in \\C \\setminus [1, \\infty)$,\n$$(c - a) \\, F(a - 1) + \\big(2a - c + (b - a) z\\big) F + a (z - 1) \\, F(a + 1) = 0,$$\n$$(a - b) \\, F = a \\, F(a + 1) - b \\, F(b + 1).$$\n\n## Remarks\n\nThe second relation in part 2 can be checked on the [[hypergeometric series|428M453CR2]] directly.\nWrite $t_n = \\frac{(a)_n (b)_n}{(c)_n \\, n!}$ for the coefficients of $F$.\nSince $a \\, (a+1)_n = (a)_n (a + n)$, the coefficient of $z^n$ in $a F(a + 1)$ is $t_n (a + n)$, and in the same way the coefficient in $b F(b + 1)$ is $t_n (b + n)$.\nTheir difference is $t_n (a - b)$, the coefficient of $(a - b) F$.\nThe other relations follow by similar coefficient comparisons, and they extend from $\\abs{z} \u003c 1$ to the cut plane by analytic continuation.\nThere are $\\binom{6}{2} = 15$ such relations, and Gauss listed them all.\n\nExample: for $a = 1$, $b = 2$ the second relation gives $-F(1, 2; c; z) = F(2, 2; c; z) - 2 F(1, 3; c; z)$, a relation between three explicit series.\n\nBy combining contiguous relations, for generic parameters the function $F(a + m, b + n; c + l; z)$ for integers $m, n, l$ is a combination of $F$ and one contiguous function, with coefficients rational in $a, b, c, z$.\n\n## Explanations\n\n### L1 human\n\nChanging one parameter of a hypergeometric function by $1$ gives a closely related function.\nGauss found that the original function and any two of its six neighbours are tied by a simple linear equation.\nThis is like the three-term recurrence of orthogonal polynomials, and it lets one step through parameters one unit at a time.\n\n### L2 human\n\nContiguous relations give recurrences for orthogonal polynomials that are terminating ${}_2F_1$ series, such as Jacobi and Legendre polynomials.\nThey lead to Gauss's continued fraction for ratios of hypergeometric functions.\nThey also show that the vector $(F, F')$ at shifted parameters is obtained from the original one by a matrix with rational entries.\nThis is the starting point of the theory of hypergeometric difference equations.\n",
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