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  "body": "\n## Statement\n\nLet $a$ and $b$ be complex numbers with $b \\notin \\set{0, -1, -2, \\dots}$, and let $M$ be [[Kummer's confluent hypergeometric function|6AZ4ZBN1M8]].\nThen for every $z \\in \\C$,\n$$M(a, b, z) = e^{z}\\, M(b - a, b, -z).$$\n\n## Remarks\n\nProof idea: put $w(z) = e^z v(-z)$ into [[Kummer's equation|NT917KFR57]] with parameters $a, b$.\nA direct computation shows that $w$ is a solution exactly when $v$ solves Kummer's equation with parameters $b - a, b$.\nSo $e^z M(b - a, b, -z)$ is a solution of the first equation that is analytic at $0$ with value $1$.\nFor $b \\notin \\set{0, -1, -2, \\dots}$ there is only one such solution, namely $M(a, b, z)$.\nIndeed, the other exponent at $0$ is $1 - b$, and a solution with that exponent cannot be analytic with value $1$ there.\nExample: for $a = b$, the left side is $e^z$ and the right side is $e^z M(0, b, -z) = e^z \\cdot 1$.\nExample: for $a = 1$, $b = 2$, the right side is $e^z \\frac{e^{-z} - 1}{-z} = \\frac{e^z - 1}{z} = M(1, 2, z)$.\n\n## Explanations\n\n### L1 human\n\nKummer's transformation says that changing $z$ to $-z$ in Kummer's function costs only a factor $e^z$ and a change of the first parameter from $a$ to $b - a$.\nIt is the confluent analogue of the Pfaff and Euler transformations of the Gauss hypergeometric function.\nFor instance it turns the trivial fact $M(0, b, z) = 1$ into $M(b, b, z) = e^z$.\n\n### L2 human\n\nThe identity is often used to move between large positive and large negative arguments, since one side may be much easier to estimate than the other.\nIt also shows that if $a$ or $b - a$ is a nonpositive integer, then $M(a, b, z)$ is a polynomial or $e^z$ times a polynomial.\nThe Tricomi function satisfies a companion identity, $U(a, b, z) = z^{1 - b} U(a - b + 1, 2 - b, z)$, often called Kummer's second transformation.\n",
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