Improved bounds for (kn)! and the factorial polynomial

Transcription

1 Divulgaciones Matemáticas Vol 11 No 003), pp Improved bounds for kn)! and the factorial polynomial Cotas mejoradas para kn)! y el polinomio factorial Daniel A Morales danoltab@ulave) Facultad de Ciencias, Universidad de Los Andes, Apartado Postal A1, La Hechicera Mérida 5101, Venezuela Abstract In this article, improved bounds for kn)! and the factorial polynomial are obtained using Kober s inequality and Lagrange s identity Key words and phrases: factorial, factorial polynomial, Kober s inequality, Lagrange s inequality Resumen En este artículo se obtienen cotas mejoradas para kn)! y el polinomio factorial usando la desigualdad de Kober y la identidad de Lagrange Palabras y frases clave: factorial, polinomio factorial, desigualdad de Kober, identidad de Lagrange 1 Introduction Factorials are ubiquitous in mathematics and the natural sciences Thus, the evaluation of factorials is necessary in many areas Unfortunately, there is no formula for the factorials of n This fact was already recognised by Euler who, refering to factorials, stated that its general term cannot be exhibited algebraically [3] However, approximations and bounds for n! are known On one hand, we have the famous Stirling s approximation given by n! πnn/e) n for large n On the other hand, recently, bounds for n!, n)!,,kn)! and the factorial polynomial have been found [,] using the arithmetic-geometric Received 003/05/08 Revised 003/11/04 Accepted 003/11/17 MSC 000): Primary D05, D07, 05A10

2 154 Daniel A Morales inequality AGI) Thus, the problem of finding better bounds and approximations for the factorial is an interesting line of inquiry In this note, we show that improved bounds for the factorial polynomial can be found using the Kober inequality [5] together with the Lagrange identity [1] The Lagrange identity can be stated as follows [1]: Let {a 1, a,, a N } and {b 1, b,, b N } be any two sets of real numbers; then N ) k=1 a N ) N ) kb k = k=1 a k k=1 b k 1 k<j N a kb j a j b k ) The Kober inequality can be stated as follows [5]: Let {a 1, a,, a N } be any set of positive numbers with arithmetic and geometric means M a and M g, respectively; then NM a M g ) N j<k aj ) a k NN 1)Ma M g ) This last inequality is less known than the AGI, however it is known that it gives sharper bounds than the AGI [1] A new improved bound for kn)! is found as a particular case of the bound for the factorial polynomial The factorial polynomial also known as falling factorial, binomial polynomial, or lower factorial) is defined by x 0) = 1, x n) = xx 1)x ) x n 1) [4] Results Theorem 1 ) x 1 n)x n3 4n 5n n/ 1 x n) ) x 1 n)x 3n 7n n/ 1 Proof The Kober inequality, applied to the set {x, x 1),, x n 1)}, with x n and a i = x i 1) gives nm a M g ) j<kk j) nn 1)M a M g ), 1) where and M a = 1 n n x i 1) = x 1 n)x n 3 n 1 i=1 n ) 1/n n /n M g = x i 1) = x i 1)) = i=1 i=1 x n)) /n Divulgaciones Matemáticas Vol 11 No 003), pp

3 Improved bounds for kn)! and the factorial polynomial 155 Now, from the Lagrange identity with a i = i and b i = 1 we find k j) = n n 1)n 1) j<k n n 1) 4 = n 1)n n 1) 1 Finally, substitution of the last three expressions into 1) gives the result Corollary 1 ) n/ ) n/ ) 1 5n n ) n! ) 1 3n ) Proof Set x = n and use n n) = n! Corollary kn) 1 n)kn n3 4n ) n/ 5n [k 1) n)!] kn)! 1 ) n/ kn) 1 n)kn 3n 7n [k 1) n)!] 1 Proof Set x = kn and use the property kn) n) = kn)! k1)n)! Since the lower bounds are only valid for small values of n n 5) we will concentrate in what follows in the upper bounds Theorem kn)! k1 [ k i)n 1 n ) ]) n/ n [0 0k 1)n 515 4k 5k )n k 1k 4k 3 )n 3 1 1k 104k 4k 3 80k 4 )n 4 ] ) nk/4 ) kn Proof From Corollary, kn)! 1n kn ) ) 1n n/ k 1 1)n 1n ) 1 kn 1n k )n 1n ) n/ k 1)n 1n ) 1 ) ) n/ 1 [k 1) n)!] ) n/ [k ) n)!] ) n/ [k 3) n)!] ) 1 ) 1 ) n/ Divulgaciones Matemáticas Vol 11 No 003), pp

4 15 Daniel A Morales kn ) ) 1n n/ k ) ) 1 1)n 1n n/ 1 k ) ) )n 1n n/ 1 k ) ) 1 k)n 1n n/ [k k) n)!] 1 ) n/ kn ) ) 1n n/ k ) 1 1)n 1n 1 k ) ) )n 1n n/ n 1)3n )) n/ [ kn ) ) 1n k ) ) 1 1)n 1n 1 k )n 1n The set 1 ) 1 ) 1 1 n 1)3n ))] n/ { kn ) 1n 1, k 1)n 1n ) 1,, 1 n 1)3n )} contains k terms Applying Kober s theorem again [ k ) ] with a i = i)n 1n 1 gives km a M g ) k i<j [ k i)n 1 n ) k j)n 1 n ) ] kk 1)M a M g ), 3) where and M a = 1 k = M g = k1 [ k i)n 1 n ) ] n [ 0 01 k)n 515 1k 4k )n k 4k 3k 3 )n k 48k 4 )n 4] k1 [ k i)n 1 n ) ]) /k n 1 1 Now, using the Lagrange identity with a i = ) k i)n 1n and bi = 1 we obtain Divulgaciones Matemáticas Vol 11 No 003), pp

5 Improved bounds for kn)! and the factorial polynomial 157 k i<j [ k i)n 1 n ) k j)n 1 n ) ] = k1 k k i)n 1 n ) [ 4 k1 k i)n 1 n ) ] = k k 1)n [ 15 30kn 4n 1k n ] Substitution of the last expression into 3) yields ) Corollary 3 n)! Proof Set k = Corollary 4 3n)! Proof Set k = 3 [ n 1n [ n 1n [ 3n 1n ) 1 ) 1 ) ] n/ 1 ] n/ [ n 1n ] n/ [ n 1n ) ] n/ 1 ) ] n/ 1 What remains to be proven is that our result is indeed sharper than the previous one The result of [] can be rewritten concisely as Then, we have k1 Theorem 3 : kn)! [ k1 [ k i)n 1n ) ] n k i 1))n 1 4) ) ] ) n/ 1 [ k1 ki1)) ) ] n Proof : It is only necessary to show that the ith term of the right hand side of 4) is indeed greater than the ith term of the right hand side of the first inequality of ) We will proceed by reductio ad absurdum and suppose the contrary, that is, ki1)) ) k i)n 1n 1 After some simplifications on the last expression we arrive at 0, which is a contradiction since n 0 Divulgaciones Matemáticas Vol 11 No 003), pp

6 158 Daniel A Morales 3 Examples In order to compare with previous estimates, we evaluate bounds for 9!, 10! and 1! By Corollary 3 we obtain which is a better bound than ) 5/ ) 5/ ! obtained from Corollary 3 of [] By Corollary 3 we get which a better bound than 100! 59) 50/ 4) 50/ ) obtained from Corollary 3 of [] By Corollary 4 we have 191 9! 3 which a better bound than ) 3/ 74 3 ) 3/ ) 3/ found by Corollary 4 of [] Finally, by Corollary 4 we obtain 59 1! which is a better bound than ) 51 ) ) found by Corollary 4 of [] We believe that other types of bounds, along the lines of [] but using Kober s inequality instead of the AGI could be found for the factorial of n Divulgaciones Matemáticas Vol 11 No 003), pp

7 Improved bounds for kn)! and the factorial polynomial 159 References [1] Beckenbach, E F, Bellman, R, Inequalities, Springer, Berlin, 195 [] Edwards, P, Mahmood, M, Sharper bounds for the factorial of n, Math Gaz 85 July 001), [3] Euler, L, On trascendental progressions that is, these whose general terms cannot be given algebraically, Opera Omnia, I 14, 1-4 Translated from the Latin by S G Langton, 1999 This paper can be found at wwwacusd edu/~langton/ [4] Graham, R L, Knuth, D E, Patashnik, O, Concrete Mathematics: A Foundation for Computer Science, Addison-Wesley, Reading, MA, 1994, p 48 [5] Kober, H, On the arithmetic and geometric means and on Hölder s inequality, Proc Am Math Soc ), [] Mahmood, M, Edwards, P, Singh, S, Bounds for kn)! and the factorial polynomial, Math Gaz 83 March 1999), Divulgaciones Matemáticas Vol 11 No 003), pp