Applicable Analysis and Discrete Mathematics available online at

Transcription

1 Applicable Analysis and Discete Mathematics available online at Appl. Anal. Discete Math. 5 (2011, doi: /aadm d POLYNOMIALS RELATED TO HARMONIC NUMBERS AND EVALUATION OF HARMONIC NUMBER SERIES II Ayhan Dil, Veli Kut In this pape we focus on -geometic polynomials, -exponential polynomials and thei hamonic vesions. We show that hamonic vesions of these polynomials and thei genealizations ae useful fo obtaining closed foms of some seies elated to hamonic numbes. 1. INTRODUCTION In [15] the concept of hamonic-geometic polynomials and hamonic-exponential polynomials ae intoduced and hypehamonic genealizations of these polynomials and numbes ae obtained. Futhemoe, it was shown that these polynomials ae quite useful fo obtaining closed foms of some seies elated to hamonic numbes. In this pape, we extend this analysis to -vesions of these polynomials and numbes. Boyadzhiev [6] has pesented and discussed the following tansfomation fomula: (1 n=0 g (n (0 f(nx n = n=0 f (n (0 =0 x g ( (x { n whee f, g ae appopiate functions and ae Stiling numbes of the second } ind Mathematics Subject Classification. 11B73, 11B75, 11B83. Keywods and Phases. Exponential numbes and polynomials, geometic numbes and polynomials, Stiling numbes, hamonic and hypehamonic numbes. 212

2 Polynomials elated to hamonic numbes and evaluation One of the pincipal objectives of the pesent pape is to give closed foms of some seies elated to hamonic numbes as well. To this end, we give a useful genealization of (1 which contains -Stiling numbes of the second ind: (2 n= g (n (0 ( n! n f (nx n = n= f (n (0 =0 { n } x g ( (x, whee f (x denotes the Maclauin seies of f(x without the fist tems. Based on fomula (2 we intoduce the concept of -geometic and -exponential polynomials and numbes. We obtain explicit elations between the - vesions and the classical vesions of these polynomials and numbes. Besides, we pesent hamonic (and hypehamonic vesions of -geometic and -exponential polynomials and numbes as well. On the othe hand, fomula(2 and hamonic -geometic polynomials enable us to obtain closed foms of the following seies n= ( n!n m H n x n, whee m and ae integes such that m and H n is the n-th patial sum of the hamonic seies. In the est of this section we discuss some impotant notions. Stiling numbes of the fist and second ind [ n Stiling numbes of the fist ind and Stiling numbes of the second ind ] { n ae quite impotant in combinatoics [4, 5, 11, 21]. Fo integes n 0; } [ n epesents the numbe of pemutations of n elements with exactly cycles ] { n and epesents the numbe of ways to patition a set with n elements into } disjoint, nonempty subsets [11]. We note that fo n 1, the following identity holds fo Stiling numbes of the second ind. (3 = { n 1 Thee is a cetain genealization of these numbes, namely -Stiling numbes [8], which ae simila to the weighted Stiling numbes [9, 10]. Repesentations and combinatoial meanings of these numbes ae as follows [8]: -Stiling numbes of the fist ind; [ ] n = The numbe of pemutations of the set {1,2,...,n} with cycles, such that the numbes 1,2,..., ae in sepaate cycles, }.

3 214 Ayhan Dil, Veli Kut -Stiling numbes of the second ind; = The numbe of patitions of the set {1,2,...,n} into non-empty disjoint subsets, such that the numbes 1,2,..., ae in sepaate subsets. In paticula, = 0 gives the classical Stiling numbes. The -Stiling numbes of the second ind satisfy the same ecuence elation as (3, except fo the initial conditions, i.e. [8]. { } { } n n = 0, n <, = δ,, n =, (4 } { n = , n >. Exponential polynomials and numbes Exponential polynomials (o single vaiable Bell polynomials φ n (x ae used in [2, 7, 16, 21] as follows: φ n (x := =0 x. The well nown exponential numbes (o Bell numbes ae obtained by setting x = 1 in φ n (x i.e., [3, 11, 12]. φ n := φ n (1 = =0. In [14] the authos obtained new poofs of some fundamental popeties of the exponential polynomials and numbes using Eule-Seidel method as: (5 φ n+1 (x = x =0 ( n φ (x and φ n+1 = =0 ( n φ. Recently, Mező [18] has defined the -Bell polynomials and numbes as: B n, (x = =0 + x and B n, = + =0 +, + espectively. The -exponential polynomials and numbes which we discuss in the pesent pape ae slightly diffeent than the -Bell polynomials and numbes in [18].

4 Polynomials elated to hamonic numbes and evaluation Geometic polynomials and numbes Geometic polynomials ae used in [6, 22, 23] as follows: (6 F n (x := =0!x. In paticula, x = 1 in (6 we get geometic numbes (o odeed Bell numbes F n [6, 23, 24] as: F n := F n (1 = =0!. Moeove, these numbes ae called Fubini numbes (this explains F and pefeential aangements as well. Boyadzhiev [6] intoduced the geneal geometic polynomials as (7 F n, (x = 1 Γ( =0 Γ( +x, whee Re( > 0. In section 5 we deal with the geneal geometic polynomials. Exponential and geometic polynomials ae connected by the following integal elation [6] (8 F n (x = 0 φ n (xλe λ dλ. In [14], the authos also obtained some fundamental popeties of the geometic polynomials and numbes using Eule-Seidel method as: F n+1 (x = x d [ Fn (x+xf n (x ] and F n = dx n 1 =0 ( n F. By means of -Stiling numbes, Mező and Nyul[20] intoduced -geometic polynomials and numbes (o -Fubini o odeed -Bell polynomials and numbes ae as follows: + + F n, (x = ( +! x and F n, = ( +!, + + =0 espectively. In this pape, the -geometic polynomials come up natually in an application of the genealized tansfomation fomula as well. Note that ou definition of -geometic polynomials (23 slightly diffes fom that of Mező and Nyul [20]. =0

5 216 Ayhan Dil, Veli Kut Hamonic and Hypehamonic numbes The n-th hamonic numbe is the n-th patial sum of the hamonic seies: 1 H n :=, whee H 0 = 0. Fo an intege α > 1, let H (α n := =1 =1 H (α 1, with H n (1 := H n being the n-th hypehamonic numbe of ode α [4, 12]. These numbes can be expessed in tems of binomial coefficients and odinay hamonic numbes as: [4, 12, 19]: n=1 H (α n = ( n+α 1 α 1 (H n+α 1 H α 1. The well-nown geneating functions of the hamonic and hypehamonic numbes ae given by H n x n = ln( and H n (α x n = ln( ( α, espectively [13]. The following elations connect hamonic and hypehamonic numbes with the Stiling and -Stiling numbes of the fist ind [4]: [ ] +1 (9 =!H, 2 and (10!H ( = n=1 [ ] n GENERALIZATION OF THE TRANSFORMATION FORMULA In this section we fist mention Boyadzhiev s Theoem 4.1 in [6] and give a useful genealization of it. As a esult of this genealization we intoduce - geometic polynomials and numbes. Suppose we ae given an entie function f and a function g, analytic in a egion containing the annulus K = {x C : < x < R}, whee 0 < < R. Hence these functions have the following seies expansions: f(x = p n x n and g(x = q n x n. n=0 n=

6 Polynomials elated to hamonic numbes and evaluation Now we ae eady to state Boyadzhiev s theoem. Theoem 1. [6] Let the functions f and g be descibed as above. If the seies conveges absolutely on K, then (11 holds fo all x K. n= n= q n f(nx n = m=0 q n f(nx n 2.1. Genealization of the opeato (xd The opeato (xd is defined as: m { } m p m x g ( (x =0 (xdf(x := xf (x, whee f is the fist deivative of the function f. Stiling numbes of the second ind appea in the fomula (11 due to the opeato (xd. Ou aim is to obtain a moe geneal fomula than (11 which contains -Stiling numbes of the second ind instead of Stiling numbes of the second ind. Accodingly, fist we genealize the opeato (xd. Fo any m-times diffeentiable function f we have [6], (12 (xd m f (x = m =0 { } m x f ( (x. This fact can be easily poven with induction on m by the help of (3. Ou fist aim is to genealize the opeato (xd. Late we use this genealization to obtain -geometic and -exponential polynomials and numbes. Definition 2. Let f be a function which is at least m-times diffeentiable and be a nonnegative intege. We define (xd as { (xd m 0, m < f(x := (xd m x f ( (x, m. Fom Definition 2 and the ecuence elation (4, using induction on m, we can pove that m { } m (13 (xd m f(x = x f ( (x, whee m. =0

7 218 Ayhan Dil, Veli Kut Equation (13 is a genealization of equation (12 since setting = 0 in (13 gives equation (12. If n is an intege and m, then it follows fom (13 (14 (xd m x n = n m ( n!x n Genealization of the tansfomation fomula We now give ou main theoem that is a genealization of Theoem 1. Theoem 3. Let f(x be an entie function and g(x be an analytic function on the annulus K = {x C,s < x < S}, whee 0 s < S. Suppose that thei powe seies given as f(x = p m x m and g(x = q n x n, m=0 and f (x denotes the powe seies n= p m x m. If the seies m= (15 n= q n ( n! n f (nx n conveges absolutely on K whee is a nonnegative intege, then n= holds fo all x K. q n ( n! n f (nx n = m { } m p m x g ( (x Poof. By consideing the powe seies expansion of g(x with (13 and (14, we have (16 n= q n ( n m= n m!x n = m =0 =0 { } m x g ( (x whee m and ae intege such that m 0. If we multiply both sides of equation (16 by p m and sum on m fom to infinity we get n= q n ( n! n p m n m x n = m= m { } m p m x g ( (x, m= since (15 is conveges absolutely on K. This completes the poof. =0

8 Polynomials elated to hamonic numbes and evaluation Coollay 4. Let g be an analytic function on the dis D = {x C,0 x < S} then g (n ( (0 n! (17 n f (nx n f (n (0 = x g ( (x. n= Most of the esults in the subsequent sections depend on Coollay 4. n= Rema 5. The paticula case = 0 in the Theoem 3 efes to the Theoem 4.1 of Boyadzhiev [6]. Theefoe fom now on we conside the case EXPONENTIAL POLYNOMIALS AND NUMBERS, -GEOMETRIC POLYNOMIALS AND NUMBERS Stiling numbes of the fist and second ind ae notable in many banches of mathematics, especially in combinatoics, computational mathematics and compute science [1, 5, 11, 12, 17]. The impotance of the exponential polynomials and numbes ae substantial because of thei diect connection with Stiling numbes. -Stiling numbes [8] ae one of the eputable genealizations of Stiling numbes. Theefoe, it is meaningful to intoduce the concepts of the -exponential and -geometic polynomials and numbes as follows exponential polynomials and numbes (18 n= =0 Fist, we conside g(x = e x in equation (17. Hence we get ( n! n f (n xn = f (n (0 ex x. The finite sum on the RHS is a genealization of exponential polynomials. We call these polynomials the -exponential polynomials in notation φ n (x: (19 φ n (x := x. n= =0 Asin theclassicalcase, -exponential numbes canbedefined bysettingx = 1 in (19, i.e., φ n :=. =0 Now we give an explicit fomula which connects -exponential polynomials with the classical exponential polynomials. This fomula also allows us to calculate φ n (x easily. Poposition 6. We have (20 φ n+ (x = x n whee n and ae nonnegative integes. =0 =0 ( n n φ (x,

9 220 Ayhan Dil, Veli Kut Poof. Let m be an intege such that m 0 and we put f(x = x m in (18. Then we get the following equality: ( n! (21 φ m (xe x = nm x n. n= The ight hand side of (21 can be witten as x n=0 (n+ m xn = x m =0 Using the definition of the opeato (xd we obtain Fom (12, we have m x =0 m x e x ( m =0 ( m ( m m m (xd e x. m φ (x. Compaision of the LHS and the RHS completes the poof. n=0 n xn. It is easy to see that equation (20 is a genealization of equation (5. As analogue of (20 appeas in the pape of Mező [18]. Rema 7. As a coollay of Poposition 6, a simila elation can be given between classical exponential numbes and -exponential numbes as: ( n φ n+ = n φ. = geometic polynomials and numbes (22 By consideing g(x = 1 in equation (17 we get ( n! n f (nx n = 1 f (n (0 { } ( n x!. n= n= We call the finite sum of the RHS as -geometic polynomials and indicate them with F n (x. Hence (23 F n (x :=!x. =0 =0 =0 We define -geometic numbes by specializing x = 1 in (23 as F n :=!.

10 Polynomials elated to hamonic numbes and evaluation The following poposition gives an explicit elation between -geometic polynomials and genealized geometic polynomials which wee given by equation (7. Poposition 8. Fo any nonnegative integes n and we have (24 F n+ (x = x! =0 ( n n F,+1 (x Poof. Let m be a nonnegative intege such that m. By setting f (x = x m in (22 we get the following cuious fomula: (25 Reaanging RHS of (25 gives ( 1 x F m = x! m =0 ( m n= m n=0 We can wite this by means of (xd opeato as x! m =0 ( m ( n!n m x n. ( n+ n x n. m (xd 1 ( +1. Consideing the fact that (equation (3.26 in [6] ( (xd 1 ( +1 = 1 x ( +1F,+1 completes the poof. Rema 9. Letting x = 1/2 in (25, we obtain anothe expession fo F n: ( n n m (26 F m =! 2 n 1 n= Rema 10. A simila esult between numbes is as follows. ( n (27 F n+ =! n F,+1 =0 =0 Fom (24 and (27, we have the following elations fo classical geometic polynomials and numbes as a coollay: ( ( n n F n+1(x = x F,2 (x and F n+1 = F,2. =0

11 222 Ayhan Dil, Veli Kut 4. HARMONIC -GEOMETRIC POLYNOMIALS AND NUMBERS, HARMONIC -EXPONENTIAL POLYNOMIALS AND NUMBERS We intoduced concepts of hamonic-geometic and hamonic-exponential polynomials and numbes in [15]. In this section we follow simila appoach as in [15] to investigate hamonic -geometic and hamonic -exponential polynomials and numbes Hamonic -geometic polynomials and numbes We conside the geneating function of hamonic numbes as the function g in the tansfomation fomula (17. Fom [15] we have (28 g ( (x =!( H ln( ( +1 and g ( (0 =!H. Using Theoem 3 we state the following tansfomation fomula fo hamonic numbes. Poposition 11. Let be a nonnegative intege and let f be an entie function. Then we have (29 n= ( n! H n n f (nx n = 1 ln( n= f (n (0 =0 { n f (n (0 n= =0 } ( x!. { } ( n x!h Poof. Employing (28 in (17 gives the statement. Second pat of the RHS of equation (29 contains -geometic polynomials which aefamilia to us fom the pevioussection. But the fist pat contains a new family of polynomials which is a genealization of hamonic-geometic polynomials [15]. We call them the hamonic -geometic polynomials and denote them as Fn h (x. Thus (30 F h n(x := =0!H x. Hamonic -geometic numbes can be defined by setting x = 1 in (30, i.e., F h n := =0!H.

12 Polynomials elated to hamonic numbes and evaluation (31 Hence with this notation we state the fomula (29 simply as ( n! H n n f (nx n n= { = 1 n= f (n (0 F h n ( ( } x x F n ln(. In the following coollay we obtain closed foms of some seies elated to hamonic numbes and binomial coefficients. Coollay 12. ( n (32!n m H n x n = 1 { n= F h m whee m and ae integes such that m 0. Poof. It follows by setting f (x = x m in equation (31. ( ( } x x F m ln(, Rema 13. Fomula (32 allows us to calculate closed foms of seveal hamonic numbe seies. The cases = 0 and = 1 in (32 has been analyzed in [15] aleady. (33 The case = 2 gives n m 1 (n 1H n x n n=2 = 1 { 2F h m ( ( } x x 2 F m ln(. Hence some seies and thei closed foms that we get fom (33 ae as follows. Fo m = 2 we have n(n 1H n x n = x2{ 3 2ln( } ( 3. n=2 Fo m = 3 we have n 2 (n 1H n x n = x2{ 6+5x (4+2xln( } ( 4, n=2 and so on. The case = 3 gives (34 n m 2 (n 1(n 2H n x n n=3 = 1 { 3F h m ( ( } x x 3 F m ln(.

13 224 Ayhan Dil, Veli Kut Hence some seies and thei closed foms that we get fom (34 ae as follows: Fo m = 3 we have n(n 1(n 2H n x n = x3{ 11 6ln( } ( 4. n=3 Fo m = 4 we have n 2 (n 1(n 2H n x n = x3{ 33+17x (18+6xln( } ( 5, n=3 and so on. Rema 14. In the following poposition and fom now on we use the notation s+1 n to indicate the following multiple type sums: s+1 s+1 =0 s=0 2 1 =0 Now we give a summation fomula fo the multiple seies. Poposition 15. (35 ( n ( ( n+s! m H s n= =0 ( = 1! m n= s+1 n { ( 1 x = ( s+2 Fm h. x n 1 H 1 F m ( x x n whee m, and s ae nonnegative integes such that m. } ln( Poof. Multiplying both sides of equation (32 with the Newton binomial seies and consideing that n ( =0 ( n+s s! m H = s+1 n ( 1!1 m H 1 we get the statement.

14 Polynomials elated to hamonic numbes and evaluation By setting = 2 and s = 0 in fomula (35 we can give the following applications: Fo m = 2 we have ( n ( 1H x n = x2{ 3 2ln( } ( 4. and so on. n=2 =2 Fo m = 3 we have ( n 2 ( 1H x n = x2{ 6+5x (4+2xln( } ( 5, n=2 =2 Rema 16. Using (9 we can state Fn(x h and Fn h in tems of Stiling numbes of the second ind and Stiling numbes of the fist ind as { } [ ] { } [ ] Fn h n +1 (x = x and Fn h n +1 =. 2 2 = Hamonic -exponential polynomials and numbes Beaing in mind the similaity of exponential and geometic polynomials and using the definition of hamonic exponential polynomials and numbes, we give the following definition. Definition 17. Fo nonnegative integes n and, hamonic -exponential polynomials and hamonic -exponential numbes ae defined espectively as φ h n (x := n =0 =0 H x and φ h n := n H. =0 Rema 18. Definition 17 enables us to extend the elation (8 as (36 F h n(x = 0 φ h n (xλe λ dλ. 5. HYPERHARMONIC -GEOMETRIC POLYNOMIALS AND NUMBERS, HYPERHARMONIC -EXPONENTIAL POLYNOMIALS AND NUMBERS We now conside hypehamonic numbes and thei tansfomations. In this way we can genealize almost all esults of [15] and in the pevious sections of the pesent pape.

15 226 Ayhan Dil, Veli Kut 5.1. Hypehamonic -geometic polynomials and numbes Simila to the pevious section, let us conside the function g in the tansfomation fomula (17 as the geneating function of the hypehamonic numbes. Fom [15] we have (37 g ( (x = Γ( +α Γ(α 1 ( H+α 1 ( α+ H α 1 ln( and (38 g ( (0 =!H (α. Now we give a tansfomation fomula fo hypehamonic numbes. Poposition 19. Fo integes 0 and α 1 we have (39 n= = ( n H (α n 1 ( α! n f (nx n f (n (0 n= ln( ( α n= =0 f (n (0 { } ( n!h (α x 1 Γ( +α Γ(α =0 ( x. Poof. Consideation (37 and (38 in (17 give the statement. The fist pat of the RHS is a genealization of hamonic -geometic polynomials which contains hypehamonic numbes instead of hamonic numbes. We call these polynomials the hypehamonic -geometic polynomials and denote them as Fn,α h (x. Thus (40 F h n,α(x = =0!H (α x. The second pat of the RHS of (39 contains a genealization of the, geneal geometic polynomials. We call these polynomials the geneal -geometic polynomials and denote them as F n,α (x. Hence (41 F n,α (x = 1 Γ(α =0 Γ( +αx.

16 Polynomials elated to hamonic numbes and evaluation (42 Using these notations we can state (39 simply as n= = ( n H (α n 1 ( α! n f(nxn f (n [ (0 n= F h n,α ( ( ] x x F n,α ln(. Rema 20. Putting x = 1 in (40 we get hypehamonic -geometic numbes as { } Fn,α h n =!H (α, =0 and putting x = 1 in (41 gives geneal -geometic numbes as { } F n,α = 1 n Γ( +α. Γ(α =0 Using the following coollay of Poposition 19 we obtain closed foms of some seies elated to hypehamonic numbes and binomial coefficients. Coollay 21. (43 n= = ( n!n m H n (α xn [ 1 ( α F h m,α ( ( ] x x F m,α ln(. Poof. Fo a positive integes m, setting f (x = x m in (42 gives (43. Rema 22. Putting the values of, m and α in (43, one can get closed foms of seveal hypehamonic numbes seies. Now we extend the fomula (35 to hypehamonic numbe seies. Poposition 23. Let m,,s and α be nonnegative integes such that and m. Then we have ( n ( ( n+s (44! m H (α x n s n= =0 ( = 1!1 m H (α 1 x n n= s+1 n { ( ( } 1 x x = ( α+s+1 Fm h F m ln(.

17 228 Ayhan Dil, Veli Kut Poof. Multiplying both sides of equation (43 with the Newton binomial seies gives the statement. Rema 24. Paticula values of, m, s and α in (44 gives closed foms of seveal mutiplicative hypehamonic numbes seies. Rema 25. Using (10 we get an altenative expession of hypehamonic -geometic polynomials and numbes as { } [ ] { } [ ] Fn,α(x h n +α = x, Fn,α h n +α =. α+1 α+1 = Hypehamonic -exponential polynomials and numbes α Definition 26. Fo positive integes α and, hypehamonic -exponential polynomials ae defined as φ h n,α (x = n =0 =0 H (α x. Hence hypehamonic -exponential numbes ae defined as φ h n,α = =0 Rema 27. We extend the elation (36 as H (α. α F h n,α(x = 0 φ h n,α (xλe λ dλ. Acnowledgements. The autho geatly appeciates the efeees valuable comments and suggestions. Reseach of the authos ae suppoted by Adeniz Univesity Scientific Reseach Poject Unit. REFERENCES 1. M. Abamowitz, I. Stegun: Handboo of Mathematical Functions with Fomulas, Gaphs, and Mathematical Tables, 9th pinting. New Yo: Dove, p. 824, E. T. Bell: Exponential polynomials. Ann. of Math., 35 (2 (1934, E. T. Bell: Exponential numbes. Ame. Math. Monthly, 41 (1934, A. T. Benjamin, D. Gaeble, R. Gaeble: A combinatoial appoach to hypehamonic numbes, Integes: Electon. J. Combin. Numbe Theoy, 3 (2003, 1 9 #A A. T. Benjamin, J. J. Quinn: Poofs that Really Count: The At of Combinatoial Poof. MAA, 2003.

18 Polynomials elated to hamonic numbes and evaluation K. N. Boyadzhiev: A Seies tansfomation fomula and elated polynomials. Int. J. Math. Math. Sci. 2005, 23 (2005, K. N. Boyadzhiev: Exponential Polynomials, Stiling Numbes and Evaluation of some Gamma Integals. Abst. Appl. Anal., Volume 2009, Aticle ID A. Z Bode: The -Stiling numbes, Discete Math., 49 (1984, L. Calitz: Weighted Stiling numbes of the fist and second ind-i. Fibonacci Quat., 18 (1980, L. Calitz: Weighted Stiling numbes of the fist and second ind-ii. Fibonacci Quat., 18 (1980, L. Comtet: Advanced Combinatoics. The At of Finite and Infinite Expansions, Revised and enlaged edition. D. Riedel Publishing Co., Dodecht, J. H. Conway, R. K. Guy: The boo of numbes. New Yo, Spinge-Velag, A. Dil, I. Mező: A Symmetic Algoithm fo Hypehamonic and Fibonacci Numbes. Appl. Math. Comput. 206 (2008, A. Dil, V. Kut: Investigating geometic exponential polynomials with Eule-Seidel matices. J. Intege Seq., 14, (2011 Aticle A. Dil, V. Kut: Polynomials Related to Hamonic Numbes and Evaluation of Hamonic Numbe Seies I. Submitted. Available at J. A. Gunet: Übe die Summieung de Reihen... J. Reine Angew. Math., 25 (1843, R. L. Gaham, D. E. Knuth, O. Patashni: Concete Mathematics. Addison Wesley, I. Mező: The -Bell numbes. J. Intege Seq., 14, (2011, Aticle I. Mező, A. Dil: Hypehamonic seies involving Huwitz zeta function. J. Numbe Theoy, 130 (2 (2010, I. Mező, G. Nyul: -Fubini numbes and -Euleian numbes. Manuscipt. 21. J. Riodan: Combinatoial Analysis. John Wiley, New Yo, I. J. Schwatt: An Intoduction to the Opeations with Seies. Chelsea, New Yo, S. M. Tanny: On some numbes elated to the Bell numbes. Canad. Math. Bull., 17 (5 (1974, H. S. Wilf: Geneatingfunctionology. Academic Pess, Adeniz Univesity, (Received Novembe 15, 2010 Depatment of Mathematics, (Revised June 17, Antalya Tuey s: adil@adeniz.edu.t, vut@adeniz.edu.t