EXACT COVERING SYSTEMS AND THE GAUSS-LEGENDRE MULTIPLICATION FORMULA FOR THE GAMMA FUNCTION JOHN BEEBEE. (Communicated by William Adams)

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1 proceedings of the aerican atheatical society Volue 120, Nuber 4, April 1994 EXACT COVERING SYSTEMS AND THE GAUSS-LEGENDRE MULTIPLICATION FORMULA FOR THE GAMMA FUNCTION JOHN BEEBEE (Counicated by Willia Adas) Abstract. The Gauss-Legendre ultiplication forula for the gaa function is (2K)(-')/2w1/2-zr(OTz) = r(z)t(z + Jj) -T(z + &=i). Let {a, (od bj) : 1 < i < } be an exact covering syste with standardized offsets. Then T(z/bx) fr T{{z + ai)lbi) U bix-*"»l}2b7*/bt(ai/bi)- Conversely, if the above identity holds, then {a, (od bj) : 1 < / < } is an exact covering syste with standardized offsets. The Gauss-Legendre ultiplication forula is a special case of this identity. Let Zab he the arithetic progression (AP) {x : x = a + nb, n e Z}. Another notation for this arithetic progression is a (od b). A finite collection of disjoint AP's C = {Za.b.: I < i < } is called an exact covering syste (or exact cover) if each integer belongs to exactly one AP Za.b.. We usually assue that the offsets a, have been standardized so that 0 < a, < bj. A consequence of the fact that C is an exact cover with standardized offsets is that there is one and only one offset that is zero, and we assue it is always «i. Another property of exact covers is that YlT=i V^' = 1 If the offsets are standardized, Y%Lx ail i = ( - l)/2, by Theore 1 of Fraenkel [1]. Let M = lc{6,}. If each of the integers {0, I,..., M 1} is covered by C, then all the integers are covered by C. The collection Z = {Za.b. : 1 < / < oo} is an infinite exact cover if each integer belongs to exactly one AP. There are two classes of infinite exact covers. A saturated cover has JZ^i ^l i = 1 An unsaturated cover has In [2] I proved that... (nz\ -fr sinn((aj - z)/bi) s(nz) = fe, sin I -= J I I-^-.- " \bx J fi s(naj/bj) if and only if C = {Za.b. : 1 < i < } is an exact cover. A special case of this Received by the editors April 22, 1992 and, in revised for, July 26, Matheatics Subject Classification. Priary 11B25, 33A Aerican Matheatical Society /94 $1.00+ $.25 per page

2 1062 JOHN BEEBEE is the well-known identity / \ -i-l / \ ( n\ ( (~ l)n\ sin(z) = 2 sin(z)sinl z H-J -sini z In [1] Fraenkel proves that Raabe's identity for the Bernoulli polynoials, Bn(z) = "-x^bn(z) + Bn(^z + ^ Bn^ + ^^)), can be generalized to exact covers. (See also Beebee [3].) This last identity is an additive analogy to the Gauss-Legendre ultiplication forula, (1) (2nY-x^2x/2-zT(z) = T(z)t(z + V iyz + ^ ^]. \ ) \ ) Evidently, these identities belong to a class which can be generalized to exact covers. For exaple, the referee has noinated the g-gaa function to this class (see [4]). The definition of this class and the coon characteristics of the functions need to be deterined. Theore 1. If T(z) = g(z)t[ =l T((z + a^/bi) where g has no zeros at the nonpositive integers and is defined for all coplex z, then the set ofap's C = {Za.bi : 1 < i < } is an exact cover with standardized offsets; and conversely if C is an exact cover with standardized offsets, then r,,_t(z/bi){\t((z + ai)/bt) () {)'bx^'bxf}2br/bt(ai/bi)' Proof. Suppose T(z) = g(z) T[ =1 T((z + aij/bi). The gaa function has the nonpositive integers for its only poles, and these poles have order 1. Then -a, is a pole of the function on the right and hence of the function on the left. Thus -a, is a nonpositive integer, so #, is a nonnegative integer. If n is a nonnegative integer, then -a, - nbj is a pole on the right and hence on the left. Thus a, + nbj is a nonnegative integer. Thus bi is a positive integer. If n is a nonnegative integer, -n is a pole of order 1 on the left, and hence (-n + a^/bi is a pole for precisely one / on the right and thus is a nonpositive integer: (-«+ ai)/bj = -. Hence n = a, + bj for each nonnegative n, so each nonnegative integer belongs to exactly one AP Za.b.. For finite, this eans C is an exact cover. (There are infinite saturated systes of disjoint AP's that cover the nonnegative integers but not the integers. See Exaple 3 below.) Zero is a pole of the left side, so (0 + a,)/&, is a pole on the right for soe i and hence a,/z>, is a nonpositive integer for soe i. But <z,/6, > 0, so a, = 0 for soe i. Suppose a, > 6,. Then bi - a, < 0. Hence bi - aj is a pole on the left. But this iplies it is a pole on the right. Hence (bj - a; + af)/bj = -n, for soe nonnegative n. If j = i, this iplies 1 is a pole of the gaa function, so j ^ i. Thus a, - bi = aj + nbj, which is a contradiction, because a, - Z>, cannot belong to two AP's in the exact cover. Hence 0 < a, < bi, so C has standardized offsets. Now suppose C is an exact cover with ai = 0, 0 < a, < b,. The referee suggested the following derivation of (2). It is siilar to the derivation of (1) in Rainville [5] or Marsden [6]. Let (z) = z(z + 1) (z + n - 1) and N =

3 EXACT COVERING SYSTEMS AND GAUSS-LEGENDRE MULTIPLICATION FORMULA 1063 a ultiple of all the oduli, bi. Since C is an exact cover with standardized offsets, Thus {0, 1, 2,..., N - 1} = (J{«i! + nbi: 0 < n < N/bt - 1}. ;=i (z)n= f[(z + ai)(z + ai + bi)---(z + ai+ (-^-IJbij = f\bnibl(z + ai\ By Theore 9 of [5], (z) = T(z + n)/t(z). Thus T(z + N) -ft T((z + aj)/bi + N/bj) [> T(z) l\0i T((z + ai)/bi) By Lea 7 of [5], li _00((«- l)\nz/t(z + «)) = 1. Rearranging (3) and using this, or T(z) = H _(N-1)\N*_ nr=1 T((z + a,)/a) ^-oo n i b?/bi(n/bi - l)!(/vy6,)(z+«.)a ' r(z),- (N-l)\ (4) ti- -= li- n:, ttt = const, bf1 njl, H(z + a,-)/6/) A^ ^(«-i)/2 Tf^j fef""')/*' because XI 1/A = 1 and a(7 < = (- l)/2. Taking liz_o on the left, we see that const =(*iftr(j)) Substituting this value of the constant in (4) yields (2). Neither this or any other proof that I have found applies to infinite exact covers. Exaple 1. The Gauss-Legendre ultiplication forula is a special case of (2). Proof. It is easy to see that C = {Z,_i : I < i < } is an exact cover. In [2] I proved Lea. If C = {Za.b. : 1 < i < } is an exact cover then nat L Oi Sin7T-rbi = «r- ;'=2 2~x When we apply this lea to the exact cover C, we get But T(z)r(l - z) = n/(sinnz). flsin7r(i^i) = 2^- 1=2 x ' Hence nr( V(i- ) = f\_-_= ' im^- LJ- V ) V ) ** sin7r((/ - l)/)

4 1064 JOHN BEEBEE Butr&iWHn^ro-^.so Now substitute z for z, / - 1 for a;, and for bj in (2), and use (5): _ r(z)r(z+l/7k)---r(z 1 j (2n)(-l)/2l/2-z + (-l)/>n) Rearrangeent of this gives the Gauss-Legendre ultiplication forula (1). Exaple 2. As an exaple of (2), consider the exact cover Substituting in (2) yields r(z) = ^ ' For z = 10, this equation is C = {Zfj4, Z24, Zi6, Z36, Z56}. r(f)r(^)r(^)rr(^) 4'-z/44-^6-z/66-z/66-z/6r( )r(i)r( )r( )' ~ ' o8n (1.329)(2)(.941)(1.082)(1.329)_ 4-i-54-2.S6-i.6676-i.6676-i.667(1.772)(5.566)(1.772)(1.129)" Exaple 3. C = {Z02; Z14; Z^y,... ; Z2n-\..x,i» \ Z2n-X<2n} is an exact cover, the Grey cover, with n + 1 AP's. Substitute C into (2). Tl2\ = T{z/2) r((2"-l + z)/2") ft r((2'-'-l + z)/2') ^ > ji-zji (2»)-z/2-r((2" - l)/2") y (20-z/2T((2'-1 - l)/2')' Taking li «->oo yz> on both sides we see that the product converges and r(z/2) r(i) ft r(i/2 + (z-i)/2q ji-zii 2op(l) 11 2-'z/2T(l/2-1/2')' But the set of disjoint AP's C = {Zq2; Z14; Z$t%;...; Z2»-i_i)2»; } is not an infinite saturated exact cover, even though JZllx 1/ ; = 1 > because it does not cover -1. Thus for = 00 (2) can hold, but C is not an exact cover. The proof of Theore 1 ade use of YllLi l/ < = 1, so I speculate that if C is an unsaturated infinite exact cover, (2) does not hold. But if C is a saturated infinite exact cover, then (2) does hold. Liited nuerical experients support this conjecture. Exaple 4. Define With this notation, (2) is Y"biZ) 1?(» + )/*) iw0. k b-zlbt(a/b) (6) T(z) = Y[Tab(z).!=i

5 EXACT COVERING SYSTEMS AND GAUSS-LEGENDRE MULTIPLICATION FORMULA 1065 The function Tab(z) has about the sae relation to the AP Zab that T(z) has to the integers. (a) T(z) has its poles at the nonpositive integers. Tab(z) has its poles at the nonpositive integers in Zab. (h) For n > 1, T(l + z) = zt(z), and T(n) = n - l\. For the functions Tab(z), we have Tob(b) = l and TQb(nb) = (n-l)b-(n-2)b---(2b)(b), Tab(b) = a and Tab(nb) = (a + (n - l)b) (a + (n - 2)b) (a + 2b)(a + b)(a). Let C he an exact cover, and let z = M = lc{o,}. Then (6) is the self-evident forula *-u-((*-')*0((*-jh"»* fl (a. + (^ - l) 6,) (a, + (^ - 2) &, ) (at + b,) at. References 1. Aviezri S. Fraenkel, A characterization of exactly covering congruences, Discrete Math. 4 (1973), John Beebee, Soe trigonoetric identities related to exact covers, Proc. Aer. Math. Soc. 112 (1991), _, Bernoulli nubers and exact covering systes, Aer. Math. Monthly 99 (1992), George Gasper and Mizan Rahan, Basic hypergeoetric series, Cabridge Univ. Press, Cabridge, Earl D. Rainville, Special functions, Macillan, New York, Jerrold E. Marsden, Basic coplex analysis, Freean, San Francisco, B. Novak and S. Zna, Disjoint covering systes, Aer. Math. Monthly 81 (1974), Sheran K. Stein, Unions of arithetic sequences, Math. Ann. 138 (1958), M. A. Berger, A. Felzenbau, A. S. Fraenkel, and R. Holzan, On infinite and finite covering systes, Aer. Math. Monthly 98 (1991), Departent of Matheatical Sciences, University of Alaska, Anchorage, Alaska E-ail address: af jcbqalaska