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  "body": "\n## Definition\n\nLet $K$ be a field, and let $a_n, b_n, c_n \\in K$ for $n \\ge 1$ be given, with $a_n \\ne 0$ and $c_n \\ne 0$ for all $n$.\nA *three-term recurrence relation* is the homogeneous [[linear recurrence|8Y28JQB0Q5]]\n$$a_n\\, u_{n+1} + b_n\\, u_n + c_n\\, u_{n-1} = 0 \\qquad (n \\ge 1)$$\nfor a [[sequence|0G9EKM0E15]] $(u_n)_{n \\ge 0}$ in $K$.\n\n## Remarks\n\nExample: the [[Chebyshev polynomials|M6EAR1JHPP]] satisfy $T_{n+1}(x) - 2x\\, T_n(x) + T_{n-1}(x) = 0$ in $\\R[x]$, here read in the field $\\R(x)$ of rational functions with $a_n = 1$, $b_n = -2x$, $c_n = 1$.\nCheck for $n = 1$: $T_2(x) = 2x^2 - 1 = 2x \\cdot x - 1$.\nNon-example: $u_{n+1} = u_n + u_{n-1} + u_{n-2}$ links four terms, so it is not a three-term recurrence.\nThe condition $a_n \\ne 0$ lets one solve for $u_{n+1}$, and $c_n \\ne 0$ makes the order exactly two; so the solutions form a $K$-vector space of dimension $2$, fixed by $u_0$ and $u_1$.\nA common normalized form is $u_{n+1} = (x - \\alpha_n) u_n - \\beta_n u_{n-1}$, which is the [[three-term recurrence for orthogonal polynomials|YRYWCP9TYV]].\n",
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