= m 1. sin π( ai z ) )

Transcription

1 EXACT COVERING SYSTEMS AND THE GAUSS-LEGENDRE MULTIPLICATION FORMULA FOR THE GAMMA FUNCTION John Beeee Unversty of Alaska Anchorage July The Gauss-Legendre ultplcaton forula for the gaa functon s π 1 1 = Let{a od : 1 } e an exact coverng syste wth standarded offsets. Then = 1 +a. a Conversely, f the aove dentty holds, then {a od : 1 } s an exact coverng syste wth standarded offsets. The Gauss-Legendre ultplcaton forula s a specal case of ths dentty. Let Z a e the arthetc progresson AP {x : x = a + n, n Z}. Another notaton for ths arthetc progresson s aod. A fnte collecton of dsjont AP s C = {Z a : 1 } s called an exact coverng syste or exact cover f each nteger elongs to exactly one AP Z a. We usually assue that the offsets a have een standarded so that 0 a <. A consequence of the fact that C s an exact cover wth standarded offsets s that there s one and only one offset that s ero, and we assue t s always a 1. Another property of exact covers s that =1 1 = 1. If the offsets are standarded, =1 a, y Theore 1 of Fraenkel [1]. Let M = lc{ }. If each of the ntegers {0, 1,..., M 1} s covered y C, then all the ntegers are covered y C. The collecton Z = {Z a : 1 < } s an nfnte exact cover f each nteger elongs to exactly one AP. There are two classes of nfnte exact covers. A saturated cover has =1 1 = 1. An unsaturated cover has =1 1 < 1. In [] I proved that snπ = sn π 1991 Matheatcs Suject Classfcaton. 11B5,33A15. sn π a sn πa = 1 1 Typeset y AMS-TEX

2 JOHN BEEBEE f and only f C = {Z a the well known dentty : 1 } s an exact cover. A specal case of ths s sn = 1 sn sn + π 1π... sn +. In [1] A.S. Fraenkel proves that Raae s dentty for the Bernoull polynoals, B n = n 1 B n + B n B n + 1, can e generaled to exact covers. See also Beeee [3]. Ths last dentty s an addtve analogy to the Gauss-Legendre ultplcaton forula, 1 π 1 1 = Evdently, these denttes elong to a class whch can e generaled to exact covers. For exaple, the referee has nonated the q-gaa functon to ths class See [4].. The defnton of ths class and the coon characterstcs of the functons need to e deterned. Theore 1. If = g =1 +a where g has no eros at the nonpostve ntegers, and s defned for all coplex, then the set of AP s C = {Z a : 1 } s an exact cover wth standarded offsets; and conversely f C s an exact cover wth standarded offsets, then = 1 +a a. Proof. Suppose = g =1 +a. The gaa functon has the nonpostve ntegers for ts only poles, and these poles have order 1. Then a s a pole of the functon on the rght and hence of the functon on the left. Thus a s a nonpostve nteger, so a s a nonnegatve nteger. If n s a nonnegatve nteger, then a n s a pole on the rght, hence on the left. Thus a + n s a nonnegatve nteger. Thus s a postve nteger. If n s a nonnegatve nteger, -n s a pole of order 1 on the left, and hence n+a s a pole for precsely one on the n+a rght, and thus s a nonpostve nteger: =. Hence n = a + for each nonnegatve n, so each nonnegatve nteger elongs to exactly one AP Z a. For fnte, ths eans C s an exact cover. There are nfnte saturated systes of dsjont AP s that cover the nonnegatve ntegers ut not the ntegers. See exaple 3 elow. Zero s a pole of the left sde, so 0+a s a pole on the rght for soe and hence a s a nonpostve nteger for soe. But a 0, so a = 0 for soe. Suppose a. Then a 0. Hence a s a pole on the left. But ths ples t s a pole on the rght. Hence a+aj j = n, for soe nonnegatve n. If j =, ths ples 1 s a pole of the gaa functon, so j. Thus a = a j + n j,

3 EXACT COVERING SYSTEMS AND THE GAUSS-LEGENDRE MULTIPLICATION FORMULA FOR THE GAMMA FUNCTION3 whch s a contradcton, ecause a cannot elong to two AP s n the exact cover. Hence 0 a <, so C has standarded offsets. Now suppose C s an exact cover wth a 1 = 0, 0 a <. The referee suggested the followng dervaton of. It s slar to the dervaton of 1 n Ranvlle [5] or Marsden []. Let n = n 1 and N = a ultple of all the odul,. Snce C s an exact cover wth standarded offsets Thus =1 {0, 1,,..., N 1} = {a + n : 0 n N 1}. n = + a + a + + a + N 1 = By Theore 9 of [5], n = 3 + N =1 + n. Thus = =1 N +a + N. +a =1 N + a. N By lea 7 of [5], l n n 1!n + n = 1. Rearrangng 3 and usng ths, =1 +a = l N N 1!N! N =1 N 1, +a N or 4 =1 +a = l N N 1 N 1! N a =1 = constant, ecause 1 = 1 and a = 1. Takng l on the left, we see that 0 constant = a 1 Susttutng ths value of the constant n 4 yelds. Nether ths or any other proof I have found apples to nfnte exact covers. Exaple 1. The Gauss-Legendre ultplcaton forula s a specal case of. Proof. It s easy to see that C = {Z 1, : 1 } s an exact cover. In [] I proved.

4 4 JOHN BEEBEE Lea. If C = {Z a : 1 } s an exact cover then sn π a = 1. When we apply ths lea to the exact cover C, we get But 1 = 1 1 sn π π sn π. Hence 1 1 But 1 = 1 1, so = = 1. π sn π 1 = π = 1 π. Now susttute for, 1 for a, and for n, and use 5: = π 1 1 Rearrangeent of ths gves the Gauss-Legendre ultplcaton forula 1. Exaple. As an exaple of, consder the exact cover Susttutng n yelds = For = 10, ths equaton s 3, 880 = C = {Z 04, Z 4, Z 1, Z 3, Z 5 }

5 EXACT COVERING SYSTEMS AND THE GAUSS-LEGENDRE MULTIPLICATION FORMULA FOR THE GAMMA FUNCTION5 Exaple 3. C n = {Z 0 ; Z 14 ; Z 3,8 ;... ; Z n 1 1, n; Z n 1,n} s an exact cover, the Grey cover, wth n + 1 AP s. Susttute C n nto. = 1 n 1+ n n n n 1 n n Takng l n on oth sdes we see that the product converges and = But the set of dsjont AP s C = {Z 0 ; Z 14 ; Z 3,8 ;... ; Z n 1 1,n;... } s not an nfnte saturated exact cover, even though =1 1 = 1, ecause t does not cover 1. Thus for = can hold, ut C s not an exact cover. The proof of Theore 1 ade use of =1 1 = 1, so I speculate that f C s an unsaturated nfnte exact cover, does not hold. But f C s a saturated nfnte exact cover, then does hold. Lted nuercal experents support ths conjecture. Exaple 4. Defne Wth ths notaton, s a = = +a a f a = 0 f a 0. a. =1 The functon a has aout the sae relaton to the AP Z a that has to the ntegers. a. has ts poles at the non-postve ntegers. a has ts poles at the nonpostve ntegers n Z a.. For n 1, 1 + =, and n = n 1!. For the functons a, we have 0 = 1 a = a and and 0 n = n 1 n a n = a + n 1 a + n a + a + a. Let C e an exact cover and let = M = lc{ }. Then s the self-evdent forula M 1! = M1 1 M1 a + M 1 a + M a + a.

6 JOHN BEEBEE References 1. Aver S. Fraenkel, A characteraton of exactly coverng congruences, Dscrete Math , John Beeee, Soe Trgonoetrc Identtes Related To Exact Covers, Proceedngs of the Aercan Matheatcal Socety , John Beeee, Bernoull nuers and exact coverng systes, Aercan Matheatcal Monthly to appear. 4. George Gasper and Man Rahan, Basc Hypergeoetrc Seres, Cardge Unversty Press, Cardge, Earl D. Ranvlle, Specal Functons, The Macllan Copany, New York, Jerrold E Marsden, Basc Coplex Analyss, W.H. Freean & Copany, San Francsco, B. Novák and S. Zná, Dsjont Coverng Systes, Aercan Matheatcal Monthly , Sheran K. Sten, Unons of arthetc sequences, Math. Ann , M.A. Berger, A. Felenau, A.S. Fraenkel, and R. Holan, On Infnte and Fnte Coverng Systes, Aercan Matheatcal Monthly ,