INTEGRAL REPRESENTATION OF THE POLYNOMIAL K N (A, B, ; X)

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1 Bullein of he Marahwada Mahemaical Sociey Vol. 15, No. 2, ecember 214, Pages INTEGRAL REPRESENTATION OF THE POLYNOMIAL K N (A, B, ; X) N.M.Kavahekar 1, P.G. Andhare 2, T.B.Jagap 3 1,3 Y.C. Insiue of Science, Saara,4151,(MS), India. kavahekarnirmala@gmail.com, and dr.bjagap@gmail.com 2 R.B.N.B. College Shrirampur , is. Ahmednagar (MS), India. pandurangandandhare@gmail.com Absrac In his paper we have obained conour inegral represenaion, real inegral represenaion, infinie single inegral represenaion, finie single inegral represenaion, finie double inegral represenaion, of he polynomial K n (a, b, ; x) INTROUCTION In he usual noaion, le (λ) n denoe he Pochhammer symbol defined by Γ(λ + n) (λ) n Γ(λ) { 1, if n λ(λ + 1)...(λ + n 1), if n 1, 2, 3,... (1.1) so ha ( n) k { ( 1) k (n k)!, if k n,, if k > n. (1.2) and he Gamma funcion, Γ(z) is defined by Euler s inegral as Γ(z) The recurrence relaion for Γ(z) is given by e z 1 d, Re(z) >. (1.3) Γ(z + 1) zγ(z), Re(z) <, z 1, 2,... (1.4) 1 Keywords:Orhogonal Polynomial, Generaing funcion, inegral represenaion. Marahwada Mahemaical Sociey, Aurangabad, India, ISSN

2 3 N.M.Kavahekar, P.G. Andhare, T.B.Jagap ) The hypergeomeric funcion 2 F 1 (a, b; c; z is defined as, ) 2F 1 (a, b; c; z n The exponenial hypergeomeric funcion is defined as, ) 1F (a; ; z (a) n (b) n z n, c, 1, 2, (1.5) (c) n (a) n z n, (1.6) There are several useful sysems of polynomials of a discree variable eg. Krawchouk polynomials 3,P-75. n K n (x; p, N) 2 F 1 n, x; N; p 1, where < p < 1, (1.7) x, 1, 2,..., N and Meixner polynomials. M n (x, β, c) 2 F 1 n, x; β; 1 c 1, β >, < c < 1, x, 1, 2, (1.8) In his paper we have obained various ypes of inegral represenaions of he polynomial he K n (a, b; x) defined by K n (a, b, ; x) 2 F 1 n, a; b; 1 x K ( n) k (a) k ( 1 x (b) k k! (1.9) where < x < 1. Replacing a by x, x by p and b by N, where < p < 1 and x, 1, 2,, N he polynomial (1.9) reduces o Krawchouk polynomial (1.7). 2 SIMPLE GENERATING RELATION In his secion we obain a generaing funcion of he polynomial (1.9). We prove he following resul. Theorem 2.1 If 1 F (a; ; x) and K n (a, b; x) are he polynomials given by (1.6) and ((1.9) respecively, hen (b) n K n (a, b, ; x)() n n (1 ) b 1F a; ; (2.1)

3 INTEGRAL REPRESENTATION OF POLYNOMIAL K N (A, B, ; X) 31 Proof: Consider, (δ) n K n (a, b, ; x)() n n n k n k n k n k k k n (δ) n ( n) k (a) k ( 1 x n (b) k k! n (δ) n ( 1) k ()(a) k ( 1 x n (n k)!(b) k k! ( 1) k (δ) n+k (a) k ( 1 x n+k (b) k k! ( 1) k (δ) k (δ + k) n (a) k ( 1 x n+k (δ) k (a) k x k k!(b) k (b) k k! (δ + k) n n n (δ) k (a) k x k (1 ) (δ+k) k!(b) k (δ) k (a) k x(1 ) k 1 k!(b) k (1 ) δ k 1 (1 ) δ (δ) k (a) k k!(b) k x(1 ) k 1 (1 ) δ 2 F 1 δ, a; b; k Taking δ b, we obain (2.1) as (b) n K n (a, b, ; x)() n n (1 ) b 1F a; ; If we pu a x, x p and b N in he above polynomial hen i reduces o Krawchouk polynomial as below ( N) n K n (x; p, N)() n n (1 ) N 1F x; ; p( 1) (2.2) 3 INTEGRAL REPRESENTATION I Conour Inegral Represenaion: The conour inegral of a complex funcion is defined as follows: Le γ be a pah in he complex plane wih domain a, b and le f be a complex valued

4 32 N.M.Kavahekar, P.G. Andhare, T.B.Jagap funcion defined on he graph of γ. Then conour inegral of f along γ denoed by f is given by γ γ f b a fγ()dγ() whenever he inegral on he righ hand side exiss. Here we obain conour inegral represenaion of he polynomial (1.9). For his ake f() (1 ) b 1F a; ; We recall he Maclaurian s heorem f() f n ()() n n where f n () f() d, n 1, 2, 3, 2πi c n+1 and c is any closed conour encircling he origin. From (2.1) we have (b) n K n (a, b, ; x)() n n (1 ) b 1F a; ; f() n 1 b n (1 ) 2πi c 1 F a; ; d. n This furher implies ha K n (a, b, ; x) (n)! n 1 (1 ) b 1F a; ; d (3.1) (b) n 2πi c where he conour inegraion encircles he origin of he plane in he posiive direcion. If we replace a by x, x by p and b by N in he above polynomial (3.1) hen i reduces o he Krawchouk polynomial as follows. K n ( x; N, p) II Real Inegral Represenaion: n 1 (1 ) N 1F a; ; d (3.2) ( N) n 2πi c Here we obain he inegral represenaion of he polynomial K n (a, b, ; x) in inerval o 2π.

5 INTEGRAL REPRESENTATION OF POLYNOMIAL K N (A, B, ; X) 33 Now pu e iθ so ha d ie iθ in euaion 3.1 2π K n (a, b, ; (x)) (n)! (e iθ ) n 1 (1 e iθ ) b e 1F a; iθ ; 2πi(b) n x(1 e iθ ie iθ dθ ) (n)! 2π e inθ (1 e iθ ) b e 1F a; iθ ; 2π(b) n x(e iθ dθ 1) (n)! (a) 2π k e inθ (1 e iθ ) b e iθ kdθ 2π(b) n k! k x(e iθ 1) (n)! (a) 2π k ( 1) k x k e (k n)iθ (1 e iθ ) b k dθ 2π(b) n k! k (n)! ( 1) k x k (a) 2π k e (k n)iθ (b + k) s (e iθ ) s dθ 2π(b) n k! k s! s (n)! ( 1) k x k (a) k (b + k) 2π s e (k n+s)iθ dθ 2π(b) n k!s! k s (n)! ( 1) k (a) k (b + k) s x k 2π(b) n k! s! 2π k s cos(k n + s)θ + isin(k n + s)θdθ. (3.3) If we replace a by x, x by p and b by N in he above polynomial hen i reduces o Krawchouk polynomial as below K n ( x; N, p) (n)! 2π( N) n k s ( 1) k ( x) k ( N + k) s (p k ) k! s! 2π cos(k n + s)θ + isin(k n + s)θdθ (3.4) III Infinie Single Inegral Represenaion: Here we obain he inegral represenaion of he polynomial K n (a, b, ; x) in inerval o. Consider he equaion (1.9), K n (a, b, ; x) k k k k ( n) k (a) k ( 1 x (b) k k! ( n) k Γ(a + k)( 1 x Γ(a)(b) k k! ( n) k ( 1 x Γ(a + k ) Γ(a)(b) k k! ( n) k (x) k Γ(a + k ) Γ(a)(b) k k!

6 34 N.M.Kavahekar, P.G. Andhare, T.B.Jagap Using Γ(ρ k ) e 2 2(ρ k) d K n (a, b, ; x) 1 Γ(a) 1 Γ(a) 1 Γ(a) ( n) k (b) k k! (x) k e 2 2(a+k 1 2 ) d e 2 2a 1 ( n) ( k 2 ) kd (b) k k! x k k e 2 2a 1 1F 1 n; b; 2 x d (3.5) If we replace a by x, x by p and b by N in he above polynomial hen i reduces o Krawchouk polynomial as below K n ( x; N, p) 1 Γ( x) IV Finie Single Inegral Represenaion: e 2 2x 1 1F 1 n; N; 2 d (3.6) p Here we obain he inegral represenaion of he polynomial K n (a, b, ; x) in inerval, 1. Consider he equaion (1.9), K n (a, b, ; x) Γ(b) Γ(a)Γ(b a) We recall he resul 1, p-13,14 K n (a, b, ; x) k ( n) k (a) k ( 1 x (b) k k! ( n) k Γ(a + k)γ(b a)(x) k k β(x, y) Γ(x)Γ(y) Γ(x + y) Γ(b + k! 1 x 1 (1 ) y 1 d... (A) (3.7) Using he above resul in equaion (3.7) we have Γ(b) ( n) k x k 1 K n (a, b, ; x) a+k+1 (1 ) b a 1 d Γ(a)Γ(b a) k! k Γ(b) 1 ( n) k ( x a 1 (1 ) b a 1 d. Γ(a)Γ(b a) k! Using we ge n k (a) n y n (1 y) a 2, p-47, K n (a, b, ; x) Γ(b) Γ(a)Γ(b a) Γ(b) Γ(a)Γ(b a) 1 1 a 1 (1 ) b a 1( 1 x) nd a 1 (1 ) b a 1( x ) nd (3.8) x

7 INTEGRAL REPRESENTATION OF POLYNOMIAL K N (A, B, ; X) 35 If we replace a by x, x by p and b by N in he above polynomial, hen i reduces o he Krawchouk polynomial as below. K n ( x; N, p) Γ( N) Γ( x)γ( N + x) 1 V Finie ouble Inegral Represenaion: x 1 (1 ) N+x 1( p ) nd (3.9) p Here we obain he inegral represenaion of he polynomial K n (a, b, ; x) in erms of finie double inegral inegraion represenaion. Firs we recall he resul, if Re(a), Re(b) and Re(c), hen u a 1 v b 1 (1 u v) c 1 dudv Γ(a)Γ(b)Γ(c) (3.1) Γ(a + b + c) where is bounded by he lines u >, v > and u + v 1...4, p-275. ( n) k (a) k ( 1 x K n (a, b, ; x) (b) k k! k ( n) k Γ(b)Γ(a + k)( 1 x Γ(a)Γ(b + k! k 1 ( n) k Γ(b)Γ(a + k)(α) k Γ(α a b)( 1 x Γ(a) Γ(α+k) k Γ(b + k!γ(α a b) ( n) k (α) k Γ(b)Γ(a + k)γ(α a b)( 1 x Γ(a)Γ(α a b) Γ(b + k)γ(α + k! k ( n) k (α) k u a+k 1 v b 1 (1 u v) α a b 1 Γ(a)Γ(α a b) k!γ(b + k) k ( 1 x dudv(using(3.1)) ( n) k (α) k ( u x u a 1 v b 1 (1 u v) α a b 1 dudv Γ(a)Γ(α a b)γ(b) k!(b) k k u a 1 v b 1 (1 u v) α a b 1 2F 1 n; α; b; u dudv Γ(a)Γ(α a b)γ(b) x u a 1 v b 1 (1 u v) α a b 1 K n (α, b, ; x)dudv Γ(a)Γ(α a b)γ(b) If we replace a by x, x by p and b by N in he above polynomial, hen he value of K n ( x; N, p) (i.e. Krawchouk s polynomial)is Γ( x)γ(α + x + N)Γ( N) u x 1 v N 1 (1 u v) α+x+n 1 K n (α, N, ; p)dudv

8 36 N.M.Kavahekar, P.G. Andhare, T.B.Jagap References 1 N.N. Lebedev, Special Funcions and Their Applicaions, Prenice-Hall, Englewood Cliffs, New Jersery, (1965). 2 E.. Rainville, Special Funcions, Macmillan, New Yark; Reprened by Chelsea Publ.co., Bronx, New Yark, (1971) 3 H.M. Shrivasava, and H.L. Manocha, A Treaise on Generaing Funcions, Ellis Harwood Ld, (1984). 4 H.M. Shrivasava, and P.W. Karlsson, Muliple Gaussion Hypergeomeric Series, Halsed Press (Ellis Harwood Ld., Checheser) John Wiley and Sons, New Yark,(1985).