Yi Wang Department of Applied Mathematics, Dalian University of Technology, Dalian , China (Submitted June 2002)
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1 SELF-INVERSE SEQUENCES RELATED TO A BINOMIAL INVERSE PAIR Yi Wang Department of Applied Mathematics, Dalian University of Technology, Dalian , China (Submitted June 2002) 1 INTRODUCTION Pairs of inverse relations are very useful in the study of combinatorial identities [2] One of the classical inversion formulas is the binomial inverse pair: a n = ( 1) b ; b n = ( 1) a (1) =0 We say that a sequence {a n } of complex numbers is self-inverse or invariant if ( 1) a = a n (2) =0 and denote by S + the set of self-inverse sequences We also denote by S the set of sequences {a n } satisfying ( 1) a = a n =0 { } By the definition it is not difficult to verify that {a n } S if and only if a 0 = 0 and an+1 n+1 S + or {na n 1 } S + Recently, Sun [5] studied self-inverse sequences by using their generating functions and gave many interesting examples and results of self-inverse sequences In this paper, we explore self-inverse sequences by means of linear transformations, difference operators and the umbral calculus We obtain various characterizations of self-inverse sequences from these different approaches For S + we show that it is a vector space over the complex field and determine its dimension We also give simpler proofs to certain results of Sun It is worth noting that our results can give rise to many interesting identities Let P = Then the inverse pair (1) can be written as =0 2 LINEAR TRANSFORMATIONS 1 ( ( )) 1 1 O n ( 1) = α = P β ; β = P α, where α = (a 0, a 1, a 2, ) T and β = (b 0, b 1, b 2, ) T are two column vectors Note that the pair in (1) is self-inverse Hence, P is an involutory matrix, ie, P 2 = I, where I is the identity 46
2 matrix In what follows we identify a vector α = (a 0, a 1, a 2, ) T with the sequence {a n } Then α S + if and only if α is invariant under the linear transformation y = P x, ie, P α = α Thus, S + is precisely the null space of the matrix P I Similarly, we may show that S is the null space of the matrix P + I The following proposition is therefore immediate Proposition 21: Let α, β be two complex vectors (a) If α, β S ±, then aα + bβ S ± for arbitrary complex numbers a and b; (b) if A is a matrix satisfying AP = P A, then α S ± implies that Aα S ± Theorem 22: α S ± if and only if α = (P ± I)v for some vector v In other words, S ± is precisely the column space of the matrix P ± I Proof: Suppose that α S + Then P α = α Putting v = α/2, we have (P + I)v = (P + I)α/2 = (P α + α)/2 = (α + α)/2 = α Conversely, suppose that α = (P + I)v for some v Then P α = P (P + I)v = (P 2 + P )v = (I + P )v = α Thus, α S + The statement for S may be proved similarly Remar 23: The above discussion for the binomial inverse pair is also suitable for general self-inverse pairs: a n = A(n, )b ; b n = A(n)a, where the matrix A = (A(n, )) satisfies A 2 = I An equivalent form of Theorem 22 is the following theorem, which has been observed by Sun [5, Remar 32] Theorem 24: The sequence {a n } S ± if and only if there exists a sequence λ 0, λ 1, λ 2, of complex numbers such that a n = ( 1) λ ± λ n, n = 0, 1, 2, =0 Taing λ n = λ n in Theorem 24 and noting ( 1) λ = (1 λ) n, we obtain =0 Corollary 25: Suppose that λ is a complex number Then {(1 λ) n ± λ n } S ± Remar 26: Let {a n } is a sequence defined by { an+2 = a n+1 + ta n, n 0, a 0 = 2, a 1 = 1 Then, the Binet form of {a n } is a n = (1 λ) n + λ n, where λ is a root of the equation λ 2 λ t = 0 Thus, {a n } S + In particular, we have {L n } S +, where the L n are the Lucas numbers defined by L 0 = 2, L 1 = 1 and L n = L n 1 + L n 2 for n 2 47
3 Similarly, if a sequence {a n } satisfies the recursive relation a n+2 = a n+1 +ta n (n 0) with a 0 = 0, then {a n } S In particular, we have {F n } S, where the F n are the Fibonacci numbers defined by F 0 = 0, F 1 = 1 and F n = F n 1 + F n 2 for n 2 Now let m be a positive integer and P m the m m matrix ( ( 1) ( n )), n, = 0, 1,, m 1 Denote S m ± = {α C m : P m α = ±α} Then S m ± is the column space of the matrices P m ±I Moreover, we have the following result Theorem 27: With the above notation, we have (a) C m = S m + Sm, and (b) dim S m + = m/2 and dim Sm = m/2, where x and x denote the least integer greater than or equal to x and the greatest integer less than or equal to x respectively Proof: (a) Any vector α C m can be written in the form α = β + γ, where β = (P m + I)α/2 and γ = (P m I)α/2 Sm By Theorem 22, β S m + and γ Sm So C m = S m + + Sm On the other hand, it is clear that S m + Sm = {0} Thus, we conclude that C m = S m + Sm (b) For = 0, 1,, m 1, let u be the th column of the matrix O P m + I = ( m 1 1 ) ( m 1 ) 2 ( ) m 1 3 ( 1) m Then, u 0, u 2, u 4, are linearly independent since the position of the first non-zero element of these vectors is different The ran of matrix P m + I is therefore not less than m/2 It follows that dim S m + m/2 from Theorem 22 Similarly, we have dim Sm m/2 However, dim S m + + dim Sm = m by (a) Hence, dim S m + = m/2 and dim Sm = m/2, as claimed Remar 28: From the proof of Theorem 27(b) we see that u 0, u 2, u 4, form a basis of the vector space S + m Similarly we may determine a basis of the vector space S m 3 DIFFERENCE OPERATORS Given a function f : Z C, define a new function f : Z C by f(n) = f(n + 1) f(n) is called the forward difference operator For 1, we may iterate times to obtain the th forward difference operator, f = ( 1 f), 48
4 where 0 is the identity operator It is well-nown (see, eg, [4, p37]) that n f(0) = ( 1) n f() =0 Thus, Theorem 24 can be restated as follows Theorem 31: The sequence {a n } S + if and only if there exists a function f : Z C such that a n = ( 1) n n f(0) + f(n), n = 0, 1, 2, It is also well-nown that if f is a polynomial of degree d, then f(0) = 0 for > d (see, eg, [4, Proposition 142(a)]) Hence, we have Theorem 32: If f is a polynomial of degree d, then there exists an self-inverse sequence {a n } such that a n = f(n) when n > d Example 33: Let d be a positive integer 1 Suppose that f(n) = n d Then n f(0) = n!s(dn) where S(d, n) are the Stirling numbers of the second ind (see, eg, [4, Proposition 142(c)]) Thus, we obtain a self-inverse sequence a n = ( 1) n n!s(d, n) + n d satisfying a n = n d for n > d 2 Suppose that f(n) = ( n d) Then n f(0) = δ nd by induction, where δ nd = 1 if n = d and δ nd = 0 otherwise Thus, we obtain a self-inverse sequence a n = ( 1) n δ nd + ( n d) satisfying a n = ( n d) for n > d Theorem 34: Suppose that {a n } S + Then { a n+ } S + for 0 Proof: We use induction on The case = 0 is trivial Now assume that a n+ S + Then there exists a function f such that It follows that a n+ = ( 1) n n f(0) + f(n), n = 0, 1, 2, +1 a n+(+1) = ( a (n+1)+ ) = [( 1) n+1 n+1 f(0) + f(n + 1)] = ( 1) n+2 n+2 f(0) ( 1) n+1 n+1 f(0) + f(n + 1) = ( 1) n n+1 f(1) + f(n + 1) = ( 1) n n F (0) + F (n), where F (n) = f(n + 1) Hence { +1 a n+(+1) } S + and the proof is complete by induction In the case = 1, Theorem 34 states that {a n } S + yields {a n+2 a n+1 } S +, which is precisely the result of Corollary 31(c) in [5] Remar 35: The discussion for S + in this section is also suitable for S 49
5 4 UMBRAL CALCULUS Many results in combinatorics are often easily read, verified and expanded by means of the umbral method In this section we apply the umbral method to explore self-inverse sequences But here, we do not want to describe the umbral method (for a rigorous description, see [1] or [3]) The basic idea of the umbral method relies on the use of a notation where certain exponents can be interchanged with suffixes For example, (2) can be written symbolically as (1 a) n a n, with the understanding that the expression on the left be expanded in powers of a, and then each term a be replaced by a The symbol a is referred to as an umbra, and the symbol is used to denote symbolic or umbral equivalences, in which we put a a Thus, a sequence {a n } S + if and only if (1 a) n a n for all n Similarly, {a n } S if and only if (1 a) n a n for all n Proposition 41: Let {a n } and {b n } be two sequences of complex numbers (a) Suppose that both {a n } and {b n } are in S + or in S Then, ( 1) a n b = 0 =0 holds for odd n (b) Suppose that {a n } S and {b n } S + Then, (3) holds for even n Thus, Proof: Suppose that both {a n } and {b n } are self-inverse Then, a n (1 a) n, b n (1 b) n (a b) n [(1 a) (1 b)] n (b a) n When n is odd, we have (a b) n 0 Expanding the binomial expression yields (3) Other results may be obtained similarly Corollary 42: Suppose that {a n } S ± Then for odd n, ( 1) a a n = 0 =0 It is easy to see that the sequence {1/2 n } is self-inverse By Proposition 41, if {a n } S +, then ( 1) 2 a n = 0 (4) =0 holds for odd n, and if {a n } S, then (4) holds for even n The converse is also true Theorem 43: Let {a n } be a sequence of complex numbers (a) {a n } S + if and only if (4) holds for odd n (b) {a n } S if and only if (4) holds for even n 50
6 Proof: (a) It suffices to show sufficiency Suppose that (4) holds for odd n Then (a 1/2) n 0 for odd n Thus, (a 1/2) n (1/2 a) n for all n It follows that (1 a) n [1/2 + (1/2 a)] n [1/2 + (a 1/2)] n a n Hence, {a n } S + (b) The proof is similar to that of part (a) Finally, we apply the umbral method to give an elegant proof of the following proposition due to Sun [5, Theorem 41] Proposition 44: Suppose that {f n }, {a n } are two sequences of complex numbers (a) If {a n } S +, then =0 ( n ) ( f ( 1) n ( ) )f s a n = 0 (5) s (b) If {a n } S, then =0 ( n ) ( f + ( 1) n ( ) )f s a n = 0 s Proof: We only prove (a) The proof of (b) is similar Suppose that {a n } S + Then, (f + a) n [f + (1 a)] n [(f + 1) a] n Expanding and noting that (f + 1) ( s) fs, we obtain which yields (5) =0 f a n = ( ) ( ) n ( 1) n f s a n, s =0 Remar 45: We have seen that the Lucas sequence {L n } S + and the Fibonacci sequence {F n } S It is also easy to see that {( 1) n B n } S + where B n are the Bernoulli numbers defined by B 0 = 1, B 1 = 1/2 and n ( n =0 ) B = B n for n > 1 So, the results of this section can produce many identities about these well-nown numbers ACKNOWLEDGMENTS The author thans Professor Zhi-Wei Sun and Professor Zhi-Hong Sun for drawing his attention to self-inverse sequences REFERENCES [1] A P Guinand The Umbral Method: A Survey of Elementary Mnemonic and Manipulative Uses Amer Math Monthly 86 (1979): [2] J Riordan Combinatorial Identities New Yor: John Wiley & Sons, 1968 [3] S Roman The Umbral Calculus San Diego: Academic Press,
7 [4] R P Stanley Enumerative Combinatorics, Vol I Cambridge: Cambridge Univ Press, 1997 [5] Z H Sun Invariant Sequences Under Binomial Transformations The Fibonacci Quarterly 394 (2001): AMS Classification Numbers: 05A19, 05A10, 05A40, 15A04 52
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