A NOTE ON RAMUS IDENTITY AND ASSOCIATED RECURSION RELATIONS

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1 A NOTE ON RAMUS IDENTITY AND ASSOCIATED RECURSION RELATIONS CHARLES S. KAHANE Abstract. A system of recursion relations is used to establish Ramus identity for sums of binomial coefficients in arithmetic progression. As a by-product another recursion relation is obtained, which has also been used to derive Ramus identity. m=1 1. Introduction Consider the sums of binomial coefficients 0 (n = ( n/] n m=0, ( m and (n+j/] n j (n =, j = 1, 2,..., 1, n 1. m j In 4] Ramus obtained the following formula for these sums: j (n = 1 ωp j (1 + ω p n, j = 0, 1,..., 1, n 1, (2 where the ω p s are the th roots of unity: l=1 ω p = e ip2π/, p = 0, 1,, 1. (3 His derivation of this identity is described in 1] as well as in 5]. In 2] Konvalina and Liu gave a different proof of (2. Their proof was based on showing that for each fixed j = 0, 1,..., 1, j (n satisfied the recursion relation ( j (n + = ( 1 l 1 j (n + l ( 1 ] j (n, n 1. (4 l In this note we want to give another derivation of (2, based also on recursion relations. In our case, in contrast to (4, our recursions are a system of relations satisfied simultaneously by all the j s. Namely 0 (n = 0 (n (n 1, 1 (n = 1 (n (n 1, (5 2 (n = A( 2 (n 1 + A( (n 1, (n = A( (n 1 + A( 0 (n 1, n 2, (1 48 VOLUME 46/47, NUMBER 1

2 A NOTE ON RAMUS IDENTITY AND ASSOCIATED RECURSION RELATIONS with the initial conditions 0 (1 = 1, 1 (1 = 2 (1 = = 2 (1 = 0, A( The validity of (5 follows from Pascal s formula ( ( ( n n 1 n 1 = + p p p 1 (1 = 1. (6 ( m = 0 if in its extended form for n 2 and p any integer, with the understanding that ( m > m 1, while = 0 for < 0 and m 1. In the next section we will prove Ramus identity as a conseuence of (5 and (6. Finally, in the last section of the paper, we will show how the recursions (5 lead to Konvalina and Liu s recursion (4. 2. Proof of Ramus Identity To obtain identity (2 from the system (5 and the conditions (6, we first attempt, in accordance with the theory of difference euations 3], to find solutions of the system in the form j (n = v j λ n, j = 0, 1,..., 1, (7 for appropriate λ s and v j s. Inserting (7 into (5, we find that v j λ n = v j λ n 1 + v j+1 λ n 1, j = 0, 1,..., 2, and v λ n = v λ n 1 + v 0 λ n 1 ; which will surely be satisfied if v j λ = v j + v j+1, j = 0, 1,..., 2, and v λ = v + v 0. In matrix form the latter conditions assert that v λ v 1 v 2 = v In other words λ has to be an eigenvalue of the matrix on the right, with v 0, v 1, v 2,..., v being the components of an eigenvector corresponding to this eigenvalue. Now the eigenvalues λ are the zeros of the determinant 1 λ λ D (λ = λ 1 0 ( λ. Expanding, according to the first column, we find that v 0 v 1 v 2 v D (λ = (1 λ + ( 1. (11 FEBRUARY 2008/ (8 (9

3 THE FIBONACCI QUARTERLY Thus for λ to be a root of the determinant (10, λ 1 will have to be a th root of unity. Hence, the roots λ p, p = 0, 1,..., 1, of (10 are given by λ p = 1 + ω p, where ω p = e ip2π/, p = 0, 1,..., 1. (12 Knowing what the eigenvalues of the matrix on the right of (9 are, one can determine the components v j of the corresponding eigenvectors in accordance with the conditions (8. With λ = λ p = 1 + ω p, these conditions read (1 + ω p v j = v j + v j+1, j = 0, 1,..., 2, and (1 + ω p v = v + v 0. The first 1 conditions lead to v j+1 = ω p v j, j = 0, 1,..., 2, so that v j = ωpv j 0, j = 0, 1,..., 2. The remaining condition leads to v = ωp 1 v 0 = ωp v 0. Thus, in all cases, the components v j of the eigenvector corresponding to the eigenvalue λ p = 1 + ω p are given by where v 0 can be chosen arbitrarily. If we now form the expressions v j = ω j pv 0, j = 0, 1,..., 1, (13 A j (n = v j λ n = c p ω j p (1 + ω p n, j = 0, 1,..., 1, n 1, where the arbitrary constant c p represents v 0 in (13, it then follows from the preceding calculations that these A j (n s are solutions of the same system (5 that the j (n s satisfy. Superposing, the same is true of the expressions A j (n = c p ωp j (1 + ω p n, j = 0, 1,..., 1, n 1. (14 In view of the uniueness of the solution of the system of recursion relations (5 satisfying the initial conditions (6, to prove that the j (n s are given by the formula (2, it will be sufficient to show that by choosing the c p s in (14 all eual to 1/, the resulting expressions satisfy the initial conditions (6. In other words we need to show that 1 ωp j (1 + ω p = 1 and this follows immediately by noting that ( ω j 0 if j = 1, 2,..., 2 p + ωp j+1 = 1 if j = 0 or j = 1; ωp l = ω p l = ( 1 ω l / (1 ωl = 0 for l = 1, 2,...., 1, while as ωp l = 1 if l = 0 or l =, ωp l = if l = 0 or l =. 50 VOLUME 46/47, NUMBER 1

4 A NOTE ON RAMUS IDENTITY AND ASSOCIATED RECURSION RELATIONS 3. Konvalina and Liu s Recursion Relation Here we want to indicate how the recursion relation (4 can be derived from our approach. To this end we introduce the operator E which sends the nth term B (n of a seuence into the (n + 1 st term B (n + 1; the νth iterant of E, E ν, then sends B (n into B (n + ν, which we write as E ν B (n = B (n + ν. Next, given a polynomial P (x = ν=0 a νx ν, we define the corresponding operator P (E = ν=0 a νe ν in the obvious way as P (E B (n = a ν E ν B (n = ν=0 a ν B (n + ν. Using this notation the recursion relations (5 can be written as (1 E j (n + j+1 (n = 0 j = 0, 1,..., 2, and (1 E (n + A( 0 (n = 0, n 1. Viewing this as a system of linear euations in the unnowns j (n, j = 0, 1,..., 1, with operator coefficients, Cramer s rule, suitably interpreted 3, p. 112], is applicable; applying it, it follows that each j for n 1, where ν=0 (n, satisfies the recursion relation D (E j (n = 0, 1 E E D (E = E E. Clearly this determinant is the same as the determinant D (λ on the right of (10 with λ replaced by E. In (11 we found D (λ = (1 λ + ( 1. Hence, each j (n satisfies (1 E + ( 1 ] j (n = 0, n 1. Expanding, this is euivalent to j (n + = l=1 ( 1 l 1 ( l precisely the recursion relation (4 of Konvalina and Liu. j (n + l ( 1 ] j (n, n 1, FEBRUARY 2008/

5 THE FIBONACCI QUARTERLY References 1] V. E. Hogatt and G. L. Alexanderson, Sums of Partition Sets in Generalized Pascal Triangles I, The Fibonacci Quarterly, 14.2 (1976, ] J. Konvalina and Y.-H. Liu, Arithmetic Progression Sums of Binomial Coefficients, Appl. Math Lett., 10 (1997, ] H. Levy and F. Lessman, Finite Difference Euations, The Macmillan Company, New Yor, ] C. Ramus, Solution Générale d un Problème d Analyse Combinatoire, J. Reine Angew. Math., 11 (1834, ] E. Netto, Lehrbuch der Combinatori, 2nd ed., Teubner, Berlin, MSC2000: 11B37, 11B65 Department of Mathematics, Vanderbilt University, Nashville, Tennessee address: c.ahane@comcast.net 52 VOLUME 46/47, NUMBER 1