#A31 INTEGERS 18 (2018) A NOTE ON FINITE SUMS OF PRODUCTS OF BERNSTEIN BASIS POLYNOMIALS AND HYPERGEOMETRIC POLYNOMIALS

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1 #A31 INTEGERS 18 (2018) A NOTE ON FINITE SUMS OF PRODUCTS OF BERNSTEIN BASIS POLYNOMIALS AND HYPERGEOMETRIC POLYNOMIALS Steven P. Clar Department of Finance, University of North Carolina at Charlotte, Charlotte, North Carolina spclar@uncc.edu Received: 6/12/17, Revised: 11/30/17, Accepted: 3/21/18, Published: 3/30/18 Abstract This note considers finite sums of products of Bernstein basis polynomials and Gauss hypergeometric polynomials for which all three parameters are non-positive integers. A simple formula is derived for such sums and an interesting binomial identity is obtained as a special case. 1. Preliminaries The th Bernstein basis polynomial of degree n 2 N is defined by B,n (z) z (1 z) n, z 2 C. The set {B,n (z)} n 0 is a basis for the space of polynomials of degree at most n with complex coe cients. Since B,n (z) (z (1 z)) n 1, 0 the Bernstein basis polynomials of degree n form a partition of unity. For p, q 2 N, the generalized hypergeometric function is defined by 1X ( 1 ) pf q ( 1,..., p ; 1,..., q; z) n ( p ) n z n ( 1 ) n ( q ) n n!, z 2 C (1) where 1,..., p, 1,..., q 2 C and ( ) n is the Pochhammer symbol defined by ( 1 n 0 ( ) n ( 1) ( n 1) n > 0. The 2 F 1 case is called the Gauss hypergeometric function 1X ( ) 2F 1 (, ; ; z) n ( ) n z n, z 2 C. (2) ( ) n n!

2 INTEGERS: 18 (2018) 2 Clearly 2 F 1 is symmetric in and. The series in (2) terminates if and only if either 2 N or 2 N. In this case, the series is called a hypergeometric polynomial. If 2 N, then for m, m 2F 1 (, ; ; z) ( 1) n ( ) n z n. (3) n ( ) n If 2 N, first use symmetry of 2 F 1 in and, then apply (3). In general, if 2 N, the series in (2) does not converge, except in the case that the series terminates and > m. (See [2] for a definition of 2 F 1 explicitly including the case of negative integer values of.) We note that in the special case, the hypergeometric polynomial simplifies as 2F 1 (, ; ; z) 1 F 0 ( ; z) (1 z). (4) In the non-terminating case, Equation (4) only holds for z < 1, but in the polynomial case it is just a statement of the binomial theorem and thus holds for all z 2 C. In this note, we prove some simple, but non-trivial results on sums of products of Bernstein basis polynomials and hypergeometric polynomials. More specifically, we will establish a formula for finite sums of the form B,N (z) 2 F 1 (, m; N; '(z)), 0 for m, N 2 N. We will use the contiguous relationship 2F 1 (, ; ; z) 2 F 1 ( 1, ; 1; z) ( ) 2 F 1 (, ; 1; z) 0 (5) proved in [1]. Also see [4] for an up-to-date list of nown contiguous relations for the Gauss hypergeometric function. Since (5) holds for all z 2 C, it also holds if we replace z with '(z) for any C-valued function ', or branch of ' if it is multi-valued. 2. Main Results Theorem 1. Let N 2 N, m 2 N with m apple N. For ' a C-valued function (or a branch of a C-valued function) and z in its domain, if S m (n) B,n (z) 2 F 1 (, m; n; '(z)) 0 then S m (N) S m (m). Thus, if N 2 N, m 2 N with m apple N, the sum S m (N) depends only on m.

3 INTEGERS: 18 (2018) 3 Proof. Note that S m (N) S m (m) P N nm1 S m(n) S m (n 1). We prove that P N nm1 S m(n) S m (n 1) 0. For n > m, S m (n) (1 z)s m (n 1) z (1 z) n 2F 1 (, m; n; '(z)) (1 z) z (1 z) n 1 2F 1 (, m; (n 1); '(z)) 0 z (1 z) n 2F 1 (, m; n; '(z)) z (1 z) n 2F 1 (, m; (n 1); '(z)) 0 (1 z) n z n 2F 1 ( n, m; n; '(z)) 1 z (1 z) n 2F 1 (, m; n; '(z)) (1 z) n z (1 z) n 2F 1 (, m; (n 1); '(z)) 1 1 z n 2F 1 ( n, m; n; '(z)) z (1 z) n h n 2F 1 (, m; n; '(z)) 1 1 i 2F 1 (, m; (n 1); '(z)) z n 2F 1 ( n, m; n; '(z)) 1 z (1 z) n h n n 1 2F 1 (, m; n; '(z)) 1 1 n n 1 1 z n 2F 1 ( n, m; n; '(z)) 1 1 z (1 z) n h 1 n 2 F 1 (, m; n; '(z)) i 2F 1 (, m; (n 1); '(z)) i (n ) 2 F 1 (, m; (n 1); '(z)) z n 2F 1 ( n, m; n; '(z)) 1 1 z (1 z) n h 1 i 2 F 1 ( ( 1), m; (n 1); '(z)) 1 (6) (7)

4 INTEGERS: 18 (2018) 4 z n 2F 1 ( n, m; n; '(z)) 2 1 z 0 1 (1 z) n F 1 ( 0, m; (n 1); '(z)) (8) z n 2F 1 ( n, m; n; '(z)) z n 2F 1 ( (n 1), m; (n 1); '(z)) 1 1 z 0 1 (1 z) n F 1 ( 0, m; (n 1); '(z)) z n (1 z) m z n (1 z) m 1 1 z z 0 (1 z) n 1 0 2F 1 ( 0, m; (n 1); '(z)) (9) 0 0 zs m (n 1), 0 where (6) follows from the binomial coe n 1 and 1 cient identities 1 n 1, 1 (7) uses the contiguous relationship in (5), (8) follows from the substitution 0 1, and (9) follows from the symmetry of 2 F 1 in and, and an application of (4). This argument implies S m (n) S m (n 1), and the result follows. Corollary 1. Let N 2 N, m 2 N with m apple N. If ' is a C-valued function (or a branch of a C-valued function) and z is in its domain, then B,N (z) 2 F 1 (, m; N; '(z)) (1 z'(z)) m. (10) 0 Proof. The result follows from Theorem 1 and the binomial theorem: B,N (z) 2 F 1 (, m; N; '(z)) 0 B,m (z) 2 F 1 (, m; m; '(z)) 0 B,m (z) 1 F 0 ( 0 0 ; '(z)) m z (1 z) m (1 '(z)) 0 m (z z'(z)) (1 z) m (1 z'(z)) m.

5 INTEGERS: 18 (2018) 5 Specific identities that are special cases of Corollary 1 include, (i) (ii) (iii) (iv) B,N (z) 2 F 1 (, m; N; z) (1 z 2 ) m ; and in particular 0 B,N (i) 2 F 1 (, m; N; i) 2 m, 0 B,N (z) 2 F 1 (, m; N; 1 z) (z 2 z 1) m, 0 B,N (z) 2 F 1 0 B,N (z) 2 F 1 0, m; N; 1 z ( 1 m 0 0 m > 0,, m; N; 1 1 z m. z The following corollary establishes a binomial identity that is also an immediate consequence of Corollary 1. Corollary 2. Let N 2 N, m 2 N with m apple N. Then B,N (x) 2 F 1 0, m; N; 1 x y (y x) m, y 2 C, x 2 C \ {0}. (11) x A remarable aspect of this identity is the way that monomial terms of degree other than m in the summation in (11) combine and vanish, leaving only terms of degree m involved in the binomial expansion. For example, in the case with m 3 and N 4, 4X B,4 (x) 2 F 1 0 x 4 4 x 3 6 x 2 4 x 1, 3; 4; 1 x y x x 4 3 x 3 y 6 x 3 9 x 2 y 12 x 2 9 xy 10 x 3 y 3 3 x 3 y 3 x 2 y 2 3 x 3 12 x 2 y 6 xy 2 9 x 2 15 xy 3 y 2 9 x 6 y 3 3 x 2 y 2 xy 3 6 x 2 y 6 xy 2 y 3 3 x 2 9 xy 3 y 2 4 x 3 y 1 xy 3 3 xy 2 3 xy x y 3 3xy 2 3x 2 y x 3 (y x) 3. As another application, Corollary 1 can be used to derive the generating function for Krawtchou polynomials. For 0 < p < 1, N 2 N, and x 2 N with x apple N, the

6 INTEGERS: 18 (2018) 6 Krawtchou polynomials are defined for 2 N with apple x by K (x, p, N) 2 F 1 (see [3], for instance). In (10), let z X N N (1 t) N 2F 1 0 t B,N 2F 1 1 t x t t p, x; N; 1 p, t 1t and '(z) 1 p. Then, x; N; ' t 1 t t, x; N; ' 1 p p t 1 t t x (1 t) x. Thus 0 N 2F 1, x; N; 1 p t 1 x 1 p t (1 t) N x, p which is the generating function for Krawtchou polynomials as given in [3] Acnowledgement. I would lie to than the anonymous referee for several useful suggestions. References [1] Y.-J. Cho, T.-Y. Seo, and J.-S. Choi, A note on contiguous function relations, East Asian Math. J. 15 (1999), [2] A. Erdelyi (ed.), Higher Transcendental Functions: Vol. I: Bateman manuscript project, McGraw-Hill Boo Company, New Yor, [3] S. Karlin and J. McGregor, Ehrenfest Urn Models, J. Appl. Probab. 2 (1965), [4] M. A. Raha, A. K. Rathie, and P. Chopra, On some new contiguous relations for the Gauss hypergeometric function with applications, Comput. Math. Appl. 61 (2011),