MATHEMATICS OF COMPUTATION, VOLUME 33, NUMBER 146 APRIL 1979, PAGES 794-804 Bernoulli Related Polynomials and Numbers By Ch. A. Charalambides Abstract. The polynomials fn'x\a, b) of degree n defined by the equations Wn-l,b ■V>„(x; a, b) = -—- and A&y>n(x;a, b) = ¥>„_,(*; a, b) b" -in- 1)! where (x)_ ¡, = x'x — b)'x - 2b) ■ ■ ■ 'x - nb + b) is the generalized factorial and aa/(x) = fix + a) —fix), are the subject of this paper. A representation of these polynomials as a sum of generalized factorials is given. The coefficients, Bin, s), s = a/b, of this representation are given explicitly or by a recurrence relation. The generating functions of <P„ix; a, b) and Bin, s) are obtained. The limits of ip ix; a, b) as a -* 1, b -» 0 or a -* 0, b -» 1 and the limits of Bin, s) as s -» ± °° or i -* 0 are shown to be the Bernoulli polynomials and numbers of the first and second kind, respectively. Finally, the generalized factorial moments of a dis- crete rectangular distribution are obtained in terms of Bin, s) in a form similar to that giving its usual moments in terms of the Bernoulli numbers. 1. Definitions and General Results. Let Afl denote the difference operator de- fined by Aa/Tx) = fix + a) - fix) and (x)n b the generalized factorial of degree « de- fined by (x)„ b = x(x - b)(x - 2b) ■■ • (x - nb + b). Let <¿>„(x;a, b) be a polyno- mial of degree « satisfying the following equation 0-1) Aa*nix;a,b)= {X)n-l'b =b-» + x( * ). bn-i .(„-iV \n-l/b From this equation and since x \ / x ib\n-ljb [n it follows that Ab Aa ^„(x; a, b) = \ *>„_,(x; a, b). Performing the operation A"1 on both members of this equation and noting that the operations Aa and A6 are commutative, we obtain (1-2) ¿Wx;a, b) = vn_xix;a, b). Received December 15, 1977; revised August 21, 1978. AMS iMOS) subject classifications (1970). Primary 10A40, Secondary 62E99. Key words and phrases. Difference operator, generalized factorial, Stirling polynomials, Stirling numbers of the first and second kind, Bernoulli polynomials and numbers of the first and second kind, generating functions, probability factorial moments. © 1979 American Mathematical Society 0025-571 8/79/0000-0075/$03.75 794 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use BERNOULLI RELATED POLYNOMIALS AND NUMBERS 795 Since we are dealing with polynomials only, this solution is univocal (see for example Jordan [7, p. 102]) and the polynomials *p„(x; a, b) are completely determined by the equations (1.1) and (1.2). The expansion of the polynomial into a Newton series may be written as O-3) V„(*;a, b)= ¿ cmb~n+4 * ) , m=o \n-m/b where the coefficients are independent of the degree of the polynomial. To determine them we may proceed as follows: From (1.1) we deduce Aai/?,(x; a, b) = 1 ; and hence, i/>,(x; a, b) = x/a + c, so that c0 = 1/s, s = a/b. For n > 1 we have Aa<pn(0;a, b) = 0. On the other hand, we have from (1.3) Aavn(0;a, b)= Y cmb-"+">\Aa( * )l m=o L \n-m/bi x=o Since L "\n- m/bJx=o L\n-m/b I«- mJb\x=o \n~m we get "-1 / a \ Y cmb-n+m[ =o m=o \n-m b or s (1-4) Y cm[ " =0, s = a/b. m=o \n-m/ Starting from c„ = 1/s we may determine by aid of (1.4), step by step, the coeffi- cients cm, m = 1,2, . We note that these coefficients depend only on the ratio a/b and not explicitly on a and b. From (1.4) we obtain c, = -(s - l)/2s, c2 = (s - l)(s + l)/12s and so on. Introducing the numbers B(m, s) = m\cm, Eq. (1.3) may be written as t1"5> *n(x;a,b) = - Y { W *Mb)n-m n\ m=0 \mj or symbolically as V„(x; a, b) = -((x/b) + B( , s))n, (x/bf = (x/b)k, Bm( , s) = Ä(m, s). «! Performing the operation Aa 1 if « > 1 and since Aa</>„(0;a, b) = 0 we get the sym- bolic formula (1.6) ((s) + B( , s))n - B(n, s) = 0 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 796 CH. A. CHARALAMBIDES giving the numbers B(n, s). Starting from B(0, s) = 1/s, we may obtain 5(1, s) = ~(s- l)/2s, B(2, s) = (s- l)(s + l)/6s and so on. A second expansion of the polynomial ^„(x; a, b) into a series of generalized factorials (x)k a may be obtained as follows: Using the expansion (see [4]), (*)„-! b = L b""1 ■a-kOn ~hk, s)ix)ka, s = a/b, k=0 (1.1) may be written as Aa^„(x; a, b) =-— Y Ci" ~ 1, K s>Tfc(x)fcjfl. (« - 1)! t=o Since Aä1(*)fca=^L±LiL+A'.a 'k'a aik + 1) we obtain 1 X^ Wfc+1 a *nix;a,b)=---Y C\n-l,k,s) '°X+K. («-l)!fc=o (k + l)ak+x But <çn(0;a, b) = B(n, s)/n\ and hence K = B(n, s)/n\ so that (1.7) *„(*; a, Z>)= -— X OC«- 1, A",s)--±r. +-—. (n-l)U=o (k + l)ak+x n\ From (1.7) it follows that « , (™ - 1)! A>„(0; a, b) = )--¿- C(» - 1, m - 1, i) (n - 1)! and (1 -8) *„(0; a, 6) = <pn(a;a, ft) = B(n, s)/n\. Using (1.7), we may determine the numbers 77(«z, s) in terms of the numbers C(«, k, s). From (1.7) and the relation W*+i ,a = Z ak+ '¿>-mCKA:+ 1, m, r)(x)m b, r = b/a = 1/s, m = 0 we have *»(*;*' Ô) = '(-rT7 S Cin-l,k,s)-^— Y Cik + l,m, r)b-mix)m b (n iy. ^=0 k + i m=0 | ¿fa ») License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use BERNOULLI RELATED POLYNOMIALS AND NUMBERS 797 or n-l 1 n-l *„ix; a,b)= Y :-77, Z rT7 Cin~l,k, s)dk + l,m,r)\b m(x)mft m = 0 in-IV- k=m-l( k + 1 ) Bin, s) Comparing the last relation with (1.5), we get -1 n-l 1 Bin - m, s) = n CXn- 1, k, s)Cik + 1, m, r), rs = 1. k=m-lY k + 1 Putting m = 1 and writing « + 1 instead of « and since CTfc+ 1, 1, r) = ir)k+x, we finally obtain Hn, s) = Y ^^ < (1.9) k=o k + 1 which on using (see [3] ) the relation Cin, k, s) = ±- Y (-l)fc_m( k]- m=o \™> may be written as (1.10) Bin,s)= Y £ (-l)fc~mf, 1. k=0 m=o \k + l/\mf 2. Generating Functions. Consider first the problem of expanding a polynomial fix) of degree « into a series of polynomials <p¡ix',a, b), i= 0, 1, . ,n, and let (2.1) fix) = Y kMÍx' a>b)- !=0 Since Amip„ix; a, b) = <p„_m(x; a, b) àmf(x) = km<p0(x;a,b)+ Y k^_m(x;a,b) i=m + l = km ■b/a + Y kff>i-mix>a> *)> m = 0,1,2,... f=m + l Performing the operation A^1 on both members of this equation, we obtain Am'xfix) = km • x/a + Y ^i-m + |fo fl>*) + * /=m + l and since ^„(0; a, ¿>)= <£„(«;a, b), (2.2) fcm= [A*-VW]X=«- [ArVtoUo = [A.ArVWl^o- License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 798 CH. A. CHARALAMBIDES When fix) is not a polynomial then the series (2.1) will be infinite and considera- tions of convergence must be made. But the coefficients km will still be given by (2.2). Let us now determine the generating function f(t;x) of the polynomials <pm(x;a, b). In this case we must have km = [AaAm-x fit;x)\t=0 = f", m = 0,1,2. Since A^-^l + t)x/b = fm_1(l + t)x/b and Afl(l + tf'" = (1 + t)x'b[(l + t)s - 1], we get ,_, tu + trfb kmtn = \w = f (i + ty -1 x = 0 Therefore tii + t)x'b (2.3) Y *mix;a,b)tm = m=0 (1 +t)s-l Putting x = 0, we obtain the exponential generating function of the numbers B(m, s) f"m t (2.4) Y Bim, s) - = - m = 0 ml (1 + ty -1 The exponential generating function (2.4) may be expanded in the following way: t Us - (-iy ["d+ry-i > (1 + t)s - 1 1 (i + ty -1 h sn+xL t j 1 +- s t ] =;+z where k¡> 2, i = 1,2, ... ,n,kx + k2 +■•■+ kn = m - n. Therefore, we con- clude that m (-1)" n I j\ n Bim,s) = ml Y -717 U , > k¡>2,i = 1,2, . ,n, Y *<= « " «• „=i s"1-1 ,= iy«,-/ /=1 3. Limits. Using the relations (see [3]) lim Cin, k, s)/sk = s(«, k), lim C(«, k, s)/sn = Sin, k), S-*0 J->±oo we get from (1.9) n $(f¡ f¿\ (3.1) Um sBin,s)= Y rT7= n\bn, i->-0s->o fc=o k + 1 (-D* (3-2) lim s"" + 17i(n, s)= Y — klsin> k) = Bn -+ + O0 k=0 ^ ' 1 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use BERNOULLI RELATED POLYNOMIALS AND NUMBERS 799 where Bn and bn denote the Bernoulli numbers of the first and second kind, respec- tively (see Jordan [7, pp.