Bernoulli Related Polynomials and Numbers

MATHEMATICS OF COMPUTATION, VOLUME 33, NUMBER 146 APRIL 1979, PAGES 794-804

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

1. Definitions and General Results. Let Afl denote the difference operator defined by Aa/Tx) = fix + a) - fix) and (x)n b the generalized factorial of degree « defined by (x)„ b = x(x - b)(x - 2b) ■■ • (x - nb + b). Let <¿>„(x;a, b) be a polynomial 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

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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

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 coefficients 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

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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) =

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 »)

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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

(2.1) fix) = Y kMÍx' a>b)- !=0 Since Amip„ix; a, b) =

àmf(x) = km

= 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-

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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

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

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where Bn and bn denote the Bernoulli numbers of the first and second kind, respectively (see Jordan [7, pp. 236, 267]). Similarly from (1.7) we may obtain 1 " xfc+1 (3.3) Hm a„(*), (3.4) a^i («-l)!fc=o fc+1 "• 6->-0

where ¥>„(x) and »/'„(x) denote the Bernoulli polynomials of the first and second kind, respectively.

4. The C and the Stirling Polynomials and the Numbers 7i(«, s). It was shown in [4] that the C-number CTx, x - n, s)/^-" is a polynomial of x of degree 2«

(4.1) Cix,x-n, sW~n = V D(n, k, s - 1)1 * k=o \2n-k with the coefficients satisfying the recurrence relation

(4.2) D(n, k, s) = (2n-k + I)sD(n, k, s) + [(« - k + 1> - n]7)(n, k-l,s) with 7?(1,0, s) = s and 7)(«, ft, i) *=0 if * > M- 1. Following Nielsen (see Jordan [7, p. 224]), let us call a C-polynomial the ex- pression f4 3ï ( i a (-l)"Cfr + l,x-n,s) (4-3) c(«-x-l;s) =-, x>«, r* ■(x + d„+2

which on using (4.1) may be written as

(4.4, ,>-x-1;0- H)- ¿*+'-*"»%;!;?g-

The generating function of c„(x; s) may be obtained as follows: From (4.3) we have i-iyCjm, «-»-l,s) c„(« - «z; s) =- sm-"-x-im)n + 2

and putting n = m - x - I we get

cm_^_i(-x -!;•)- (-îy-*- • s-*cïm, x, s)(x - 1)!//«! or

X! ( dm, x, s) = i- l)m-x-x ■xsxcm_x_x i-x-1; s).

Multiplying by t and summing for m — x + 1, x + 2, . . . , we get

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[il+ty-l]X- SXtX =XSX Y i-^)m~X~lCm_x_xi-X -US)!"1. m = x+ 1

Dividing by xsxtx + x we obtain (~i)*[i-o +ty]x i Y i-Dmcmi-x-l;s)r. xsxtx + x xt ~0

Putting -t instead of / and x instead of -x - 1, we finally get

^ , ^m SX+ X-tX 1 (4.5) Y cmix; s)tm =-. m=0 (x + l)[l-(l-ty]x + X (X + I)t

From (4.4) and using (4.2) we may obtain for the polynomials cm(x; s) the recurrence

(4.6) [(s - l)(x + 1) + n)c„_,(x; s) = (n - x)cn(x; s) + (x + 2)c„(x + 1 ; s)

withc0(x;s)= (s- l)sx + x/2. If we put x = 0 into the generating function (4.5), we have

~ st 1 Y cm(0; s)tm =- „èo m i-a-ty t

or -+1 YXL (_i)m Crr,m l(0;S) •tm = t s m=i s (i + ty -1

The second member of this relation is the exponential generating function (2.4) of the numbers Bim, s); therefore, we have (-l)ms cm_,(0; s) =-— Bim, s). mi Using the relation lim^^ cT«, k, s)/s" = 5(«, k), we may obtain from (4.3)

hm cnix;s)/s"+x = Sn(x),

where 5„(x) is the Stirhng polynomial (see Jordan [7, p. 228]). Similarly, using the relations

Urn Cin, k, s)/sk = s(«, k), hm Cl«, K -s)/i-s)k = x(«, k), i->-0 s-*o

we get from (1.3)

hm c„(« - x - 1 ; s) = - Snix). i-+0

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5. Applications. From (1.1) and (1.5) we obtain

V W*^ - *W+lfo",b)+K = —J— ((x/è)+ Bi, s))n+x + K; \n jb (" + !)! and hence,

Y W I =-K(sz) + Bi >s»" +1 -Bin + l, s)], x=oa \n)b in+ 1)1 ' "'

where the a in the symbol of the sum indicates that the summation is taken for equi- distant values of x with increment a. az Using this formula, we may determine the sum 2a fix) of the function fix) a=0 which is given by its differences Abj\0)

fix)= Z b-»(X) AlfiO). „=o \njb

Performing on both members of this relation, the operation A"1,a we get

^fix) = Y Vn+ iix;«' b)AnbfiO)+K; n = 0 and finally, az °° 1 (5.2) Ya /(X) = £ }-ñ [((SZ) + 5( ' S))" + l " B{n + l ' S)] A"bm- x-0a n=0 in + 1)

As a special case of (5.1), let a = b = 1. Since 77(0, 1) = 1, 77(«, 1) = 0 if « > 0 we obtain the well-known formula 2^._0 Q) = („ + ,)• A useful formula may be obtained from (5.2) if we take fix) = (x)_m b. Since A"bix)-m,b = b"i-m)nix)-m-n,b' we get from (5-2)

az (-l)mbm Ya ix)-m,b = -¿77777-K(^) +^( . ^))m+ 1 -B{m + 1,1)1. x=0

In statistical theory the «th factorial moment about an arbitrary origin c may be defined by the equation (see e.g. Kendall and Stuart [8, p. 63])

(5.3) M(„;b)ic) = Z (*/ - c)n bHX = x¡), j where (x)n b = x(x - b) • • • (x - nb + b) is the generalized factorial of degree «. They are used almost entirely for discrete distributions or continuous distributions grouped in intervals of width b. Formula (5.1) may be used for expressing the factorial moments p[„.¡b\ = p[m-,b)(0) about c = 0 of the discrete rectangular distribution with probability function (5.4) KX = x) = 1/7V, x = a, 2a, . . . , A/a.

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From (5.3) and using (5.1), we get

(5-5) /W) = ——— [HNs) + Bi , s))n+ x -Bin + 1, s)]. v ' N(n + 1)

The factorial moments p'/„ib\(c) may be obtained from (5.5) using the formula

^■,b)ic)=t(Kiyc)k^k-,by

The factorial moment generating function Git) = 2™=0 M(„;&)*"/"! of the distribution (5.4) may be obtained from the frequency generating function Na 1 1 ^-¿"(JV + 1) *'>= LaZtX=Z x=a N N 1-f if we replace t by (1 + t/b)x/b (cf. Kendall and Stuart [8, p. 66]). We get

1 1 - (1 + t/b?" Git) = - (1 + t/b)"-, s = ab. ' N K 1 - (1 + t/b)s

The factorial cumulant generating function Kit) = 2^=0 "(«•&)'"/"' Is tnen !Pven by (1 + t/bf - 1 (1 + t/b)s - 1 Kit) = log 2-¿f--log V-£-+ Slog(l + t/b), Nst/b st/b which can be expanded by using (2.3). Writing (2.3) in the form

- . s(i + ty~x i s Y

(5-6) s Y *mia - b; a, b)tm/m = log ° + ^ ~ \ m=l St Hence,

oo i.~ m fTn Kit) = s Y {N*miNa - b; Na, b) -

K[n;b) = in~ l)lsb-"{N

For a = b = 1 it reduces to N JL /«\ (-1)"_! «(«) «»(,.51) =- E )Bim,N)iN-l)n_m +-——, n m=o\m/ n

which may be compared with the result obtained by David and Barton [5, p. 54].

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In applied statistics when moments are calculated from a numerically specified distribution which is grouped, there is present a certain amount of approximations due to the fact that the frequencies are concentrated at the midpoints of intervals. The problem of eliminating the errors due to grouping has attracted the attention of several authors (see Kendall and Stuart [8] and the references therein). Abernethy [1] and Graig [6] gave a derivation of the corrections in the moments of a discrete variable. The corrections for the cumulants were expressed by Graig in terms of the Bernoulli numbers. The corrections for the factorial cumulants can be expressed in terms of the numbers 77(«, s). Let us suppose that N consecutive values of a random variable (r.v.) are grouped in a frequency of width a. Following Graig, we write x¡ = x + e, where x¡ is the class mark of the ith frequency class, for any true value, x, of the discrete r.v. included in this frequency and e is a r.v. with probability function

P(e = ek) = -, ek=^(k-(N + l)/2), k=l,2,...,N.

If Gx.it) and Gxit) are the exponential generating functions of the calculated factorial moments p,n^ and the corrected factorial moments p',ny respectively, then

Gx.(t) = Gx(t) Y 7,(1+°efc fc=l jV or GxXt)= G(t)\-il + ifW-W« • 1"(1+5J • ' ' \_N l-(l+tf'N] Taking the logarithms of both sides and remembering that the logarithm of the factorial moment generating function is the factorial cumulant generating function, we get r (l+r)a-l (l+r)a^-l a(N-l) "1 Kxit)= KxÁt)- [logL-J-log —^-—k*l + f)J•

Using (5.6), we finally obtain for the corrections for the factorial cumulants «In)= «('»)- (»- O'«Uni" - 1;a, 1)- £ ^„(a- TV;a,A0 + ^- (- 0"1

= Hn)-\ \iia - 1) + Bi.,a))n-X-iia-N) + B{ ., a/N))" +^Vo"

Remark. The polynomials (3m(X)defined by Carlitz [2] by their exponential generating function (57> fJ-m^-7T777^

are closely related (for given s) to the numbers B(m, s). Indeed, comparing (5.7) with (2.4), we get

(5.8) B(m, s) = sm-xßm(s-x).

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Though our interest is mainly concentrated on statistical applications, it should be noted that using (5.8) and the results obtained by Carlitz we may infer for the B(m, s) some interesting arithmetic properties analogous to those of the Bernoulli numbers.

Acknowledgments. I wish to thank the referee for his valuable comments to- wards revising this paper and for bringing to my attention the paper of Carlitz [2].

Statistical Unit University of Athens Panepistemiopolis, Athens 621, Greece

1. J. R. ABERNETHY, "On the elimination of the systematic errors due to grouping," Ann. Math. Statist., v. 4, 1933, pp. 263-277. 2. L. CARLITZ, "A degenerate Staudt-Clausen theorem," Arch. Math., v. 7, 1956, pp. 28- 33. 3. CH. A. CHARALAMBIDES, "A new kind of numbers appearing in the n-fold convolu- tion of truncated binomial and negative binomial distributions," SIAM J. Appl. Math., v. 33, 1977, pp. 279-288. 4. CH. A. CHARALAMBIDES, "Some properties and applications of the differences of the generalized factorials," SIAM J. Appl. Math., v. 36, 1979. 5. F. N. DAVID & D. E. BARTON, Combinatorial Chance, Griffin, London, 1962. 6. C. C. GRAIG, "Sheppard's corrections for a discrete variable," Ann. Math. Statist., v. 7, 1936, pp. 55-61. 7. CH. JORDAN, Calculus of Finite Differences, Chelsea, New York, 1960. 8. M. G. KENDALL & A. STUART, The Advanced Theory of Statistics, Vol. 1, Hafner, New York, 1961.

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