Table for Combinatorial Numbers and AssociatedIdentities: Table 2From the seven unpublished manuscripts of H. W. GouldEdited and Compiled by Jocelyn QuaintanceMay 3, 20101 Bernoulli Numbers BnRemark 1.1 Throughout this chapter, we assume n and p are nonnegative integers. We assume xis a real or complex number. Furthermore, for any real x, we let [x] denote the floor of x.1.1 Generating Function Definition of Bn∞Xk=0Bkxkk!=xex− 1, |x| < 2π (1.1)1.2 Alternative Definition for Bnn−1Xk=0kp=1p + 1pXk=0p + 1knp+1−kBk, n ≥ 1 (1.2)1.3 Explicit Formulas of BnBp=pXj=01j + 1jXk=0(−1)kjkkp(1.3)Bn=2n + 1n+1Xk=1(−1)k−1k + 1Bn+1k,kkXj=11j, where Bnk,k=kXj=0(−1)kkj(k − j)n(1.4)1Bn= n!nXk=0(−1)kn + 1k + 1Bn+kk,k(n + k)!(1.5)1.3.1 Alternative Formulations of Equation (1.3)Bn=nXk=0(−1)kk + 1Bnk,k(1.6)Bn=nXk=0(−1)kk + 1nXj=0j − 1n − kAj,n, where Aj,n=jXk=0(−1)kn + 1k(j − k)n(1.7)Bn= (−1)nnXk=1Ak,n(−1)k−1(n + 1)nk−1, n ≥ 1 (1.8)Bn=nXk=1(−1)kAk,n(n + 1)nk, n ≥ 1 (1.9)1.4 Vandiver’s Formulas for BnRemark 1.2 The formulas in this section are the work of H. S. Vandiver. The pertinent papers are“On generalizations of the numbers of Bernoulli and Euler”, Proc. of the National Academy ofSciences, Vol. 23, 1937, pp. 555-559 (also see Proc. of the National Academy of Sciences, Vol. 25,1939, pp. 197-201), and “Explicit expressions for generalized Bernoulli numbers”, Duke Math.Journal, Vol. 8, No. 3, Sept. 1941, pp. 575-584.Bn=nXk=0(−1)kn + 1k + 11k + 1kXj=0jn(1.10)Bn=nXk=0(−1)kn + 1k + 11k + 1k+1Xj=0jn, n ≥ 1 (1.11)Bn= (−1)nn!n + 1+n−1Xk=0(−1)knk + 11k + 1kXj=0jn, n ≥ 1 (1.12)21.5 Properties of Bn1.5.1 Recursive RelationnXk=0nkBk= (−1)nBn(1.13)1.5.2 Parity PropertiesB2n=2nXk=0(−1)kk + 1B2nk,k(1.14)B2n+1=2n+1Xk=0(−1)kk + 1B2n+1k,k= 0, n ≥ 1 (1.15)B2n=2nXk=0(−1)kAk,2n(2n + 1)2nk(1.16)B2n+1=2n+1Xk=0(−1)kAk,2n+1(2n + 2)2n+1k= 0, n ≥ 1 (1.17)2nXk=02nkBk= B2n(1.18)2n+1Xk=02n + 1kBk= 0, n ≥ 1 (1.19)n−1Xk=02n2kB2k= n, n ≥ 1 (1.20)nXk=02n + 12kB2k= n +12, n ≥ 1 (1.21)3nXk=12n2k − 1B2k=12− B2n, n ≥ 1 (1.22)nXk=12n + 12k − 1B2k=12, n ≥ 1 (1.23)2nXk=0(−1)k2nkBk= B2n+ 2n, n ≥ 1 (1.24)2 Bernoulli Polynomials Bn(x)Remark 2.1 Throughout this chapter, we assume n, r, and p are nonnegative integers. We assumex, y, a, b, and t are real or complex numbers. Furthermore, for any real x, we let [x] denote thefloor of x.2.1 Definition of Bn(x)∞Xk=0tkk!Bk(x) =textet− 1, |t| < 2π (2.1)2.1.1 Relationship to BnBn= Bn(0) (2.2)2.2 Alternative Definitions of Bn(x)p−1Xk=0kn=Bn+1(p) − Bn+1n + 1, n, p ≥ 1 (2.3)Bn+1(x) = (n + 1)nXk=0xk + 1Bnk,k+ Bn+1, where Bnk,k=kXj=0(−1)kkj(k − j)n(2.4)B2n+1(x) = (2n + 1)2nXk=0xk + 1B2nk,k, n ≥ 1 (2.5)42.3 Explicit Formulas for Bn(x)Bn(x) =nXj=01j + 1jXk=0(−1)kjk(x + k)n(2.6)Bn(x) =nXk=0nkxn−kBk(2.7)2.3.1 Application of Equation (2.7)Bn−j=nnjn−1Xk=j−1Ck+1jBn−1k,k, where 1 ≤ j ≤ n, andxn=nXk=0Cnkxk(2.8)2.4 Properties of Bn(x)2.4.1 Shift PropertyBn(1 − x) = (−1)nBn(x) (2.9)Applications of Equation (2.9)Bn(1) = (−1)nBn(2.10)Bn= (−1)nn+1Xk=1(−1)k−1k2Bn+1k,k(2.11)2.4.2 Addition PropertyBn(x + y) =nXk=0nkxn−kBk(y) (2.12)Application of Equation (2.12)(−1)nBn(y − 1) =nXk=0(−1)knkBk(y) (2.13)52.4.3 Appell Derivative PropertyddxBn(x) =(nBn−1(x), n ≥ 10, n = 0(2.14)Applications of Equation (2.14)Remark 2.2 Recall that Drxf(x) is the rthderivative of f(x) with respect to x.DrxBn(ax) = r!nrarBn−r(ax) (2.15)DxBn(ax + b) = a · n · Bn−1(ax + b) (2.16)2.4.4 Integration of Bn(x)ZBn(x) dx =1n + 1Bn+1(x) + C (2.17)Applications of Equation (2.17)ZBn(ax) dx =1a(n + 1)Bn+1(ax) + C, a 6= 0 (2.18)Z10Bn(x) dx =(0, n ≥ 11, n = 0(2.19)2.4.5 Other Integrals Involving Bn(x)Z10xjBn(x) dx =(−1)nn+jnnXk=0n + jn − kBn−kk + j + 1(2.20)Z10xjBn(x) dx = (−1)nnXk=0nkBn−kk+jk(k + j + 1)(2.21)Z10xjBn(x) dx =nXk=0nkBn−kk + j + 1(2.22)Z10xBn(x) dx = (−1)n+1Bn+1n + 1(2.23)Z10Bn(x)Br(x) = (−1)r−1Br+nr+nn= (−1)n−1Br+nr+nn, r ≥ 1, n ≥ 1 (2.24)62.4.6 Convolution PropertiesnXk=0(−1)knkBk(x)Bn−k(x) = (−1)n−1(n − 1)Bn(2.25)nXk=0(−1)knkBk(x)Bn−k(x) = (1 − n)Bn(2.26)nXk=0(−1)knkBkBn−k= (1 − n)Bn(2.27)nXk=02n2kB2kB2n−2k= (1 − 2n)B2n, n ≥ 2 (2.28)n−1Xk=12n2kB2kB2n−2k= −(2n + 1)B2n, n ≥ 2 (2.29)2.4.7 A Binomial ExpansionRemark 2.3 In the following identity, we let B(x) denote the Bernoulli Polynomial, and assumeB(x)r≡rXk=0CrkBk(x). (2.30)Then,B(x) + nn=nXk=0(−1)kn + xn − k1k + 1. (2.31)Also,B + nn=1n + 1. (2.32)72.5 Formulas Involving nthDifferencesnXk=0(−1)knkBj(k) = (−1)njXk=0jkBj−kBkn,n(2.33)∆x,1Bn(x) ≡ Bn(x + 1) − Bn(x) = nxn−1, n ≥ 0, x 6= 0 (2.34)2.5.1 Applications of Equation (2.34)xn=1n + 1nXk=0n + 1kBk(x) (2.35)rXk=0(−1)kkn−1=1n (−1)rBn(r + 1) − Bn+ 2rXk=0(−1)k−1Bn(k)!, n ≥ 1 (2.36)2.6 Polynomial Expansions Involving Bn(x)Remark 2.4 Throughout this section, we assume f (x) is a polynomial of degree n, namely,f(x) =nXk=0akxk. (2.37)2.6.1 Basic Expansion FormulasRemark 2.5 The following expansion is equivalent to the formula given by Charles Jordan onPage 248 of Calculus of Finite Differences, Chelsea Publishing, New. York, 1947.f(x) =nXj=0Bj(x)nXk=jk + 1jDkf(0)(k + 1)!(2.38)f(x) =nXk=0Bk(x)Ck, where C0=Z10f(x) dx, Ck=1k!∆x,1Dk−1f(x)|x=0(2.39)82.6.2 Raabe’s TheoremRemark 2.6 The identities in this section are found on Page 252 of Charles Jordan’sCalculus of Finite Differences.Bn(x) = rn−1r−1Xk=0Bnx + kr, r ≥ 1 (2.40)Bn(rx) = rn−1r−1Xk=0Bnx +kr, r ≥ 1 (2.41)Applications of Equation (2.40)Bn= rn−1r−1Xk=0Bnkr, r ≥ 1 (2.42)Bn12= −1 −12n−1Bn(2.43)Bn13+ Bn23= −1 −13n−1Bn(2.44)2.6.3 Generalizations of Equation (2.39)Remark 2.7 Two excellent reference for the formulas found in this subsection are Konrad Knopp’sTheorie und Anwendung der unendlichen Reihen, fourth edition, Berlin, 1947, and N. E. N¨orlund’sVorlesungen¨uber Differenzenrechung, Berlin, 1924 (Chelsea Reprint, New York 1954).Let w be a nonzero real or complex number. Then,f(x + wz) =1wZx+wxf(t) dt +nXk=1wkk!Bk(z) · ∆x,wDk−1xf(x),