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**The study of Stirling numbers and Eulerian numbers by Li Jenshoo.**
*(Chinese.
English summary)*
Zbl 0537.01001

Li Shanlan (1811–1882; styled himself Renshu) was one of the most influential Chinese mathematicians of his time. (He is perhaps best known for his translations, jointly with A. Wylie, which opened the way for the introduction of Western-style mathematics to China.) ”The present paper discusses some of Li’s contributions in the field of combinatorics. In his book “Duoji Bilei” (Various summation formulas for binomial coefficients, 1867) he presented several formulas involving binomials which go back to the Song mathematician Zhu Shijie and gave many interesting generalizations of them [cf. e.g. L. Takács, Acta. Sci. Math. 34, 383–391 (1973; Zbl 0264.05004)].

The first part of the paper under review deals with Li’s proof of the identity \[ \sum_{k=0}^p \ell^p_kx^k=[x]^{p+1}, \] where the numbers \(\ell^p_k\) are defined recursively (in modern language the \(\ell^p_k\)’s are just the moduli of the Stirling numbers of the first kind). Li derives this formula from the identity \[ \sum \frac1{p!} r(r+1)\cdots(r+p-2)(mr+p-m)=\frac1{(p+1)!} n(n+1)\cdots(n+p-1)(mr+p- m+1). \]

The rest of the paper gives an account of several of Li’s results on the numbers \(L^m_k(p)\), called “Li numbers of the second kind” by the author, which coincide with the generalized Eulerian numbers \(A_{n,k-1}(p)\) (for \(p=0\) we get the ordinary Eulerian numbers \(A_{n,k-1})\). As a typical result the identity \(\sum_{r=1}^m r^n = \sum_{k=1}^n A_{n,k}\binom{m+k}{n+1}\) (and a brief sketch of Li’s proof) is given.

The first part of the paper under review deals with Li’s proof of the identity \[ \sum_{k=0}^p \ell^p_kx^k=[x]^{p+1}, \] where the numbers \(\ell^p_k\) are defined recursively (in modern language the \(\ell^p_k\)’s are just the moduli of the Stirling numbers of the first kind). Li derives this formula from the identity \[ \sum \frac1{p!} r(r+1)\cdots(r+p-2)(mr+p-m)=\frac1{(p+1)!} n(n+1)\cdots(n+p-1)(mr+p- m+1). \]

The rest of the paper gives an account of several of Li’s results on the numbers \(L^m_k(p)\), called “Li numbers of the second kind” by the author, which coincide with the generalized Eulerian numbers \(A_{n,k-1}(p)\) (for \(p=0\) we get the ordinary Eulerian numbers \(A_{n,k-1})\). As a typical result the identity \(\sum_{r=1}^m r^n = \sum_{k=1}^n A_{n,k}\binom{m+k}{n+1}\) (and a brief sketch of Li’s proof) is given.

Reviewer: W. M. Ruppert

### MSC:

01A25 | History of Chinese mathematics |

01A55 | History of mathematics in the 19th century |

11-03 | History of number theory |

05-03 | History of combinatorics |