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**Johann Faulhaber and sums of powers.**
*(English)*
Zbl 0797.11026

Summary: Early 17th-century mathematical publications of Johann Faulhaber contain some remarkable theorems, such as the fact that the \(r\)-fold summation of \(1^ m\), \(2^ m, \dots, n^ m\) is a polynomial in \(n(n+r)\) when \(m\) is a positive odd number. The present paper explores a computation-based approach by which Faulhaber may well have discovered such results, and solves a 360-year-old riddle that Faulhaber presented to his readers. It also shows that similar results hold when we express the sums in terms of central factorial powers instead of ordinary powers. Faulhaber’s coefficients can moreover be generalized to noninteger exponents, obtaining asymptotic series for \(1^ \alpha + 2^ \alpha + \cdots + n^ \alpha\) in powers of \(n^{-1} (n+1)^{-1}\).

### MSC:

11B83 | Special sequences and polynomials |

01A45 | History of mathematics in the 17th century |

11B37 | Recurrences |

30E15 | Asymptotic representations in the complex plane |

### Online Encyclopedia of Integer Sequences:

Triangle of central factorial numbers T(2*n,2*n-2*k), k >= 0, n >= 1 (in Riordan’s notation).Triangle of numerators of coefficients of Faulhaber polynomials in Knuth’s version.

Triangle of numerators of coefficients of Faulhaber polynomials used for sums of even powers.

Triangle of denominators of coefficients of Faulhaber polynomials used for sums of even powers.

Square array T(m,n) read by antidiagonals: Sum_{k=1..n} k^m.

Triangle read by rows: T(k,s) = ((2*k+1)/(2*s+1))*binomial(k+s,2*s), 0 <= s <= k.

Inverse of triangle S(n,m) defined by sequence A156290, n >= 1, 1 <= m <= n.

The Faulhaber-Knuth a(0,n) sequence.

Triangle read by rows: coefficients in the sum of odd powers as expressed by Faulhaber’s theorem, T(n, k) for n >= 1, 1 <= k <= n.

Triangle read by rows. The numerators of the coefficients of the Faulhaber polynomials. T(n,k) for n >= 0 and 0 <= k <= n.

### References:

[1] | A. W. F. Edwards, A quick route to sums of powers, Amer. Math. Monthly 93 (1986), no. 6, 451 – 455. · Zbl 0605.40004 |

[2] | Johann Faulhaber, Academia Algebrœ, Darinnen die miraculosische Inventiones zu den höchsten Cossen weiters continuirt und profitiert werden, call number QA154.8 F3 1631a f MATH at Stanford University Libraries, Johann Ulrich Schönigs, Augspurg [sic], 1631. |

[3] | Ira Gessel and University of South Alabama Problem Group, Problems and Solutions: Solutions of Elementary Problems: E3204, Amer. Math. Monthly 95 (1988), no. 10, 961 – 962. |

[4] | Ira M. Gessel and Gérard Viennot, Determinants, paths, and plane partitions, Preprint, 1989. · Zbl 0579.05004 |

[5] | Ronald L. Graham, Donald E. Knuth, and Oren Patashnik, Concrete mathematics, Addison-Wesley Publishing Company, Advanced Book Program, Reading, MA, 1989. A foundation for computer science. · Zbl 0668.00003 |

[6] | C. G. J. Jacobi, De usu legitimo formulae summatoriae Maclaurinianae, J. Reine Angew. Math. 12 (1834), 263-272. |

[7] | John Riordan, Combinatorial identities, John Wiley & Sons, Inc., New York-London-Sydney, 1968. · Zbl 0194.00502 |

[8] | L. Tits, Sur la sommation des puissances numériques, Mathesis 37 (1923), 353-355. |

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