Li Shanlan identity
In mathematics, in combinatorics, Li Shanlan identity (also called Li Shanlan's summation formula) is a certain combinatorial identity attributed to the nineteenth century Chinese mathematician Li Shanlan. Since Li Shanlan is also known as Li Renshu, this identity is also referred to as Li Renshu identity. This identity appears in the third chapter of "Duoji bilei" (垛积比类 / 垛積比類, meaning "summing finite series"), a mathematical text authored by Li Shanlan and published in 1867 as part of his collected works. A Czech mathematician Josef Kaucky published an elementary proof of the identity along with a history of  the identity in 1964. Kaucky attributed the identity to a certain Li Jen-Shu. From the account of the history of the identity, it has been ascertained that Li Jen-Shu is in fact Li Shanlan. Western scholars had been studying Chinese mathematics for its historical value; but the attribution of this identity to a nineteenth century Chinese mathematician sparked a rethink on the mathematical value of the writings of Chinese mathematicians. "In the West Li is best remembered for a combinatoric formula, known as the 'Li Renshu identity', that he derived using only traditional Chinese mathematical methods." The identity . The Li Shanlan identity states that is used to get Another application of Vandermonde's convolution yields and hence Since formula_5 is independent of "k", this can be put in the form Next, the result gives Setting "p" = "q" and replacing "j" by "k", Li's identity follows from this by replacing "n" by "n" + "p" and doing some rearrangement of terms in the resulting expression: