Generalized Power Sum and Newton-Girard Identities
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ORIGINAL PAPER
Generalized Power Sum and Newton-Girard Identities Sudip Bera1 • Sajal Kumar Mukherjee1 Received: 12 April 2020 / Revised: 8 August 2020 Ó Springer Japan KK, part of Springer Nature 2020
Abstract In this article we prove an algebraic identity which significantly generalizes the formula for sum of powers of consecutive integers involving Stirling numbers of the second kind. Also we have obtained a generalization of Newton-Girard power sum identity. Keywords Colored Newton-Girard identity Digraphs Generalized power sum identity
Mathematics Subject Classification 05A19 05A05 05C30 05C38
1 Introduction The sum of powers of consecutive integers has a long and fascinating history. Historically the first ever formula for the sum was obtained by the Swiss mathematician Jacob Bernoulli (1654–1705), who proved the following: m mþ1 1 X m m m 1 þ 2 þ þ ðn 1Þ ¼ Bk nmþ1k ; m 0; n 1; ð1Þ m þ 1 k¼0 k 0
where Bks are the famous Bernoulli numbers. There is also a surprising relationship between the sum of powers and the Stirling numbers of second kind [2]. In fact,
& Sudip Bera [email protected] Sajal Kumar Mukherjee [email protected] 1
Department of Mathematics, Indian Institute of Science, Bangalore 560 012, India
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Graphs and Combinatorics
1m þ 2m þ þ nm ¼
n nþ1 1 X Sðm; kÞk!; where m þ 1 k¼0 k þ 1
Sðm; kÞ ¼
k k m 1X ð1Þkj j : k! j¼1 j
ð2Þ
ð3Þ
In [2], the author proved (2) along with its many generalizations using the so called binomial transform. In fact, the author proved the following general statement, and obtained various power sum identities as a corollary. Lemma 1.1 (Lemma 2.1, [2]) Let c1 ; c2 ; . . .; be a sequence of complex numbers. Then for every positive integer m and any complex number a; we have m m m X X X k a k ck ¼ j!Sða; jÞ ck : ð4Þ j j¼1 k¼1 k¼j
In this article, we prove a general identity, which proves Lemma 1.1, for positive integer a as a corollary and consequently many other well known power sum identities. Before stating our result, let us fix some notations. Let ðjÞ
fxi : 1 j r; 1 i mg and fy‘ : 2 ‘ m þ 1g be two sets of variables, and P ½mðwhere ½m ¼ f1; 2; . . .; mgÞ: Define ! r X ðjÞ Y Pr P ¼ xi ; j¼1
i2P
and for any finite set Q of positive integers, the maximum element of Q is denoted by Max(Q). Then we have the following: Theorem 1.2 m X
Pr ½kykþ1 ¼
k¼1
X U½mþ1;jUj 2
0 @
X
1 ð1ÞjUjjVj1 Pr V AyMaxðUÞ :
£6¼VUnfMaxðUÞg
We call this theorem ‘‘the generalized power sum theorem’’. Note that, if we put ¼ 1 for all 1 j r; 1 i m; and y‘ ¼ c‘1 and r ¼ a in our Theorem 1.2, we
ðjÞ xi
ðjÞ
obtain Lemma 1.1 for positive integer a: In particular, if we put xi ¼ 1 ¼ y‘ for all 1 j r; 1 i m and 2 ‘ m þ 1; we obtain the classical formula for the sum of powers (2). Newton-Girard identity is a very important result, occurring in many places in algebra and combinatorics. A combinatorial proof of Newton-Girard identity was
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