ON ASYMPTOTIC FORMULA OF THE PARTITION FUNCTION p A (n)

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1 #A2 INTEGERS 15 (2015) ON ASYMPTOTIC FORMULA OF THE PARTITION FUNCTION p A (n) A David Chritopher Department of Mathematic, The American College, Tamilnadu, India davchrame@yahoocoin M Davamani Chritober Department of Mathematic, The American College, Tamilnadu, India jothichritopher@yahoocom Received: 8/8/13, Revied: 11/11/14, Accepted: 12/18/14, Publihed: 1/19/15 Abtract Let A = {a 1, a 2,, a k } be a et of k relatively prime poitive integer Let p A (n) denote the number of partition of n with part belonging to A The aim of thi note i to provide a imple proof of the following well-known aymptotic relation of p A (n): n k 1 p A (n) (a 1 a 2 a k )(k 1)! 1 Introduction and Motivation A partition of a poitive integer n i a finite nonincreaing equence of poitive integer (x 1, x 2,, x m ) uch that x 1 + x x m = n The x i are called the part of the partition Let A be a et of poitive integer The partition function p A (n) i defined a the number of partition of n with part belonging to A The generating function of p A (n) i 1X n=0 p A (n)x n = Y a2a 1 1 x a (1) with p A (0) = 1; thi generating function i valid in the interval x < 1 For gcd(a) 6= 1, we have ( n p A p A (n) = gcd(a) gcd(a) if gcd(a) n, 0 otherwie

2 INTEGERS: 15 (2015) 2 n A gcd(a) = a where the et gcd(a) o : a 2 A Thu, it i expedient to aume alway that gcd(a) = 1 The function p A (n) i more appealing when A i a finite et of relatively prime integer Throughout thi note, we aume A to be a finite et of relatively prime poitive integer T C Brown, Wun-Seng Chou and Peter J S Shiue [3] found exact formula for p A (n) when A = 2 or 3 The exact formula for p A (n) can alo be found by mean of partial fraction decompoition of it generating function (ee [6]) Gert Almkvit [1] provided the exact formula for p A (n), without the uage of partial fraction decompoition of it generating function The following aymptotic relation of p A (n) i well-known Theorem 1 Let A = {a 1, a 2,, a k } be a et of k relatively prime poitive integer Then the following aymptotic relation hold true: p A (n) n k 1 (a 1 a 2 a k ) (k 1)! (2) In 1927, E Netto [8] pioneered in providing a proof of thi theorem and ubequently, in 1972, G Polya and G Szegö [9] gave another proof; in both the proof partial fraction decompoition of the generating function wa utilized In 1942, Paul Erdo [5] proved thi reult for the cae: A = {1, 2,, k} In 1991, S Sertoz and A E Ozlük [11] found another proof by wielding the following recurrence relation: nx 1 = C n i [p A (i) p A (i (a 1 a k ))] i=n k+2 for n > (a 1 a k ) (a a k ) + k 2, where ( 1) m k 2 C m = m for 0 apple m apple k 2 0 otherwie In 2000, Melvyn B Nathanon [7] obtained an arithmetic proof The aim of thi note i to provide a new proof of thi hitorical reult The proof furnihed in thi note i baed on the fact that: the function p A (n) i a quai polynomial Definition 2 An arithmetical function f i aid to be a Quai polynomial if, f( l + r) i a polynomial in l for each r = 0, 1,, 1, where i a poitive integer greater than 1 Each polynomial f( l + r) i called a contituent polynomial of f and i called a quai period of f In 1943, E T Bell [2] found that the function p A (n) i a quai polynomial by mean of partial fraction decompoition of it generating function In 1961, E M Wright [12] reetablihed thi finding by extracting the term (1 x t ) k from the generating function of p A (n), where t = lcm(a 1,, a k ) In 2006, uing a imilar method, O J Rødeth and J A Seller [10] obtained the quai polynomial repreentation of p A (n) in binomial coe cient form

3 INTEGERS: 15 (2015) 3 2 Proof of Theorem 1 21 A Recurrence Relation Satified by p A (n) Following recurrence relation i crucial to our proof Lemma 3 Let n be a poitive integer and let a 2 A Then, we have provided a apple n p A (n) = p A (n a) + p A\{a} (n), (3) Proof Let = (x 1, x 2,, x m ) be a partition of n with part belonging to A and let a 2 A Cae (i) Aume that x i = a for ome i Then i of the form: = (x 1,, x i 1, a, x i+1,, x m ) We enumerate thi kind of partition of n; to that end, we conider the mapping: (x 1,, x i 1, a, x i+1,, x m )! (x 1,, x i 1, x i+1,, x m ), which clearly etablihe a one to one correpondence between the following et: The et of all partition of n with part belonging to A and having a a a part; The et of all partition of n a with part belonging to A By the definition, the cardinality of the latter et i p A (n a) Thu, the number of partition of n of thi type i p A (n a) Cae(ii) Aume that x i 6= a 8i = 1, 2,, m Then, it i not hard to ee that, the enumeration of uch partition i p A\{a} (n) Thu, the reult follow 22 Main Part of the Proof A the conequence of Lemma 3, we will how that: 1 The function p A (n) i a quai polynomial with a quai period a 1 a 2 a k 2 Each contituent polynomial of p A (n) i of degree k 1 3 The leading coe cient of each contituent polynomial of p A (n) i (a1a2 a k) k 2 (k 1)! At thi juncture, we note that: etablihing the above three tatement complete the proof a one can get from thee tatement that lim l!1 for each r = 0, 1,, a 1 a 2 a k thi limit p A (a 1 a 2 a k l + r) (a 1 a 2 a k l + r) k 1 = 1 (a 1 a 2 a k ) (k 1)! 1; and the targeted etimate follow readily from

4 INTEGERS: 15 (2015) 4 Now, we prove Statement 1, 2, and 3 imultaneouly by uing induction on k Suppoe that k = 2 Let A = {a 1, a 2 } with gcd(a 1, a 2 ) = 1 Then applying Lemma 3 a 1 time, we get p A (a 1 a 2 l + r) p A (a 1 a 2 (l 1) + r) = ax 1 1 i=0 p {a1}(a 1 a 2 l + r ia 2 ) (4) for each r = 0, 1,, a 1 a 2 1 Since the congruence equation a 2 x r(mod a 1 ) ha an unique olution modulo a 1 (ee [4] pp 83-84), the right ide of the equation (4) equal 1 Then replacing l by 1,, l in equation (4) and adding, we get that p A (a 1 a 2 l + r) = l + p A (r) for each r = 0, 1, 2,, a 1 a 2 1 Thu, the function p A (n) i a quai polynomial with each contituent polynomial having degree 1 and leading coe cient 1 = (a1a2)2 2 (2 1)! Aume that the reult i true when A < k for a fixed k 3 Conider a et of k poitive integer ay A = {a 1, a 2,, a k } with gcd(a 1, a 2,, a k ) = 1 Let = gcd(a 1, a 2,, a k 1 ) Then applying Lemma 3 a 1 a 2 a k 1 time again, we get p A (a 1 a k l + r) p A (a 1 a k (l 1) + r) (5) X = p {a1,,a k 1 }(a 1 a k l + r ia k ) = 0appleiapplea 1 a k 1 1; (r ia k ) X 0appleiapplea 1 a k 1 1; (r ia k ) p { a 1,, a k 1 } a1 ak 1 (a k k 2 l + q i ) + r i for each r = 0, 1,, a 1 a 2 a k 1, where r i and q i were determined from the equality r ia k = a1 a k 1 q k 1 i + r i ; here, uniquene of r i and q i and the bound 0 apple r i apple a1 a k 1 1 follow from the diviion algorithm k 1 It i well-known that the congruence equation a k x r(mod ) (6) ha a olution if and only if gcd(a k, ) r (ee [4] pp 83-84) Furthermore, in uch cae eqn(6) will have gcd(a k, ) number of mutually incongruent olution modulo Here gcd(a k, ) = 1 and hence eqn(6) ha an unique olution modulo Since gcd( a1,, a k 1 )=1, by induction aumption, it follow that the right ide of the equation (5) i a um of a1 a k 1 polynomial and each of which i of degree k 2 with leading coe cient k 1 a 1 a k 1 k 3 a k 2 (k 2)2 k (k 2)! Conequently, the

5 INTEGERS: 15 (2015) 5 right ide um of the equation (5) i a polynomial of degree k 2 Thi implie that p A (a 1 a k l+r) i a polynomial in l of degree k 1 for each r = 0, 1,, a 1 a k 1 Now, we calculate the leading coe cient of p A (a 1 a k l + r) If one denote the leading coe cient of the polynomial p A (a 1 a k l +r) by c k 1, then by the previou obervation it follow that (k 1)c k 1 = (a 1 a k ) k 2, (k 2)! which implifie to The proof i now completed c k 1 = (a 1 a k ) k 2 (k 1)! Reference [1] Gert Almkvit, Partition with part in a finite et and with part outide a finite et, Exp Math, Vol 11 (2002), No 4, [2] E T Bell, Interpolated denumerant and Lambert erie, Amer J Math 65 (1943), [3] T C Brown, Wun- Seng Chou, Peter J-S Shiue, On the partition function of a finite et, Autrala J Comb, 27 (2003), [4] David M Burton, Elementary Number theory, Allyn and Bacon, Inc, 1980 [5] P Erdö, On an elementary proof of ome aymptotic formula in the theory of partition, Ann of Math (2), 43(1942), [6] Melvyn B Nathanon, Elementary method in Number theory, Springer-Verlag, New York- Berlin- Heidelberg (2000), [7] Melvyn B Nathanon, Partition with part in a finite et, Proc Amer Math Soc 128 (2000), [8] E Netto, Le hrbuch der Combinatorik, Teubner, Leipzig, 1927 [9] G Polya and G Szegö, Aufgaben and Lehrätze au der analyi, Springer- Verlag, Berlin, 1925 Englih tranlation: Problem and Theorem in Analyi, Springer- verlag, New York, 1972 [10] Ø J Rødeth and J A Seller, Partition with part in a finite et, Int J Number Theory 02, 455 (2006) [11] S Sertoz and A E Ozlük, On the number of repreentation of an integer by a linear form, İtanb Üniv Fen Fak Mat Fiz Atron Derg, 50 (1991) [12] E M Wright, A imple proof of a known reult in partition, Amer Math Monthly 68 (1961),

Manprit Kaur and Arun Kumar

CUBIC X-SPLINE INTERPOLATORY FUNCTIONS Manprit Kaur and Arun Kumar manpreet2410@gmail.com, arun04@rediffmail.com Department of Mathematic and Computer Science, R. D. Univerity, Jabalpur, INDIA. Abtract: