गणित में, राकाह बहुपद गिउलिओ राकाह के नाम पर ऑर्थोगोनल बहुपद हैं, क्योंकि उनके ऑर्थोगोनलिटी संबंध राका गुणांक के लिए उनके ऑर्थोगोनलिटी संबंधों के बराबर हैं।
राका बहुपदों को सबसे पहले किसके द्वारा परिभाषित किया गया था Wilson (1978) और द्वारा दिया जाता है
p n ( x ( x + γ + δ + 1 ) ) = 4 F 3 [ − n n + α + β + 1 − x x + γ + δ + 1 α + 1 γ + 1 β + δ + 1 ; 1 ] . {\displaystyle p_{n}(x(x+\gamma +\delta +1))={}_{4}F_{3}\left[{\begin{matrix}-n&n+\alpha +\beta +1&-x&x+\gamma +\delta +1\\\alpha +1&\gamma +1&\beta +\delta +1\\\end{matrix}};1\right].}
ऑर्थोगोनलिटी ∑ y = 0 N R n ( x ; α , β , γ , δ ) R m ( x ; α , β , γ , δ ) γ + δ + 1 + 2 y γ + δ + 1 + y ω y = h n δ n , m , {\displaystyle \sum _{y=0}^{N}\operatorname {R} _{n}(x;\alpha ,\beta ,\gamma ,\delta )\operatorname {R} _{m}(x;\alpha ,\beta ,\gamma ,\delta ){\frac {\gamma +\delta +1+2y}{\gamma +\delta +1+y}}\omega _{y}=h_{n}\operatorname {\delta } _{n,m},} [1] कब α + 1 = − N {\displaystyle \alpha +1=-N}
कहाँ R {\displaystyle \operatorname {R} }
x = y ( y + γ + δ + 1 ) , {\displaystyle x=y(y+\gamma +\delta +1),} δ n , m {\displaystyle \operatorname {\delta } _{n,m}} क्रोनकर डेल्टा फ़ंक्शन है और भार फलन हैंω y = ( α + 1 ) y ( β + δ + 1 ) y ( γ + 1 ) y ( γ + δ + 2 ) y ( − α + γ + δ + 1 ) y ( − β + γ + 1 ) y ( δ + 1 ) y y ! , {\displaystyle \omega _{y}={\frac {(\alpha +1)_{y}(\beta +\delta +1)_{y}(\gamma +1)_{y}(\gamma +\delta +2)_{y}}{(-\alpha +\gamma +\delta +1)_{y}(-\beta +\gamma +1)_{y}(\delta +1)_{y}y!}},} और
h n = ( − β ) N ( γ + δ + 1 ) N ( − β + γ + 1 ) N ( δ + 1 ) N ( n + α + β + 1 ) n n ! ( α + β + 2 ) 2 n ( α + δ − γ + 1 ) n ( α − δ + 1 ) n ( β + 1 ) n ( α + 1 ) n ( β + δ + 1 ) n ( γ + 1 ) n , {\displaystyle h_{n}={\frac {(-\beta )_{N}(\gamma +\delta +1)_{N}}{(-\beta +\gamma +1)_{N}(\delta +1)_{N}}}{\frac {(n+\alpha +\beta +1)_{n}n!}{(\alpha +\beta +2)_{2n}}}{\frac {(\alpha +\delta -\gamma +1)_{n}(\alpha -\delta +1)_{n}(\beta +1)_{n}}{(\alpha +1)_{n}(\beta +\delta +1)_{n}(\gamma +1)_{n}}},} ( ⋅ ) n {\displaystyle (\cdot )_{n}} पोछाम्मेर सिंबल .रोड्रिग्स-टाइप फॉर्मूला ω ( x ; α , β , γ , δ ) R n ( λ ( x ) ; α , β , γ , δ ) = ( γ + δ + 1 ) n ∇ n ∇ λ ( x ) n ω ( x ; α + n , β + n , γ + n , δ ) , {\displaystyle \omega (x;\alpha ,\beta ,\gamma ,\delta )\operatorname {R} _{n}(\lambda (x);\alpha ,\beta ,\gamma ,\delta )=(\gamma +\delta +1)_{n}{\frac {\nabla ^{n}}{\nabla \lambda (x)^{n}}}\omega (x;\alpha +n,\beta +n,\gamma +n,\delta ),} [2] कहाँ ∇ {\displaystyle \nabla } पिछड़ा अंतर ऑपरेटर है,
λ ( x ) = x ( x + γ + δ + 1 ) . {\displaystyle \lambda (x)=x(x+\gamma +\delta +1).}
कार्य उत्पन्न करना के लिए तीन जनरेटिंग फ़ंक्शन हैं x ∈ { 0 , 1 , 2 , . . . , N } {\displaystyle x\in \{0,1,2,...,N\}}
कब β + δ + 1 = − N {\displaystyle \beta +\delta +1=-N\quad } γ + 1 = − N , {\displaystyle \quad \gamma +1=-N,}
2 F 1 ( − x , − x + α − γ − δ ; α + 1 ; t ) 2 F 1 ( x + β + δ + 1 , x + γ + 1 ; β + 1 ; t ) {\displaystyle {}_{2}F_{1}(-x,-x+\alpha -\gamma -\delta ;\alpha +1;t){}_{2}F_{1}(x+\beta +\delta +1,x+\gamma +1;\beta +1;t)} = ∑ n = 0 N ( β + δ + 1 ) n ( γ + 1 ) n ( β + 1 ) n n ! R n ( λ ( x ) ; α , β , γ , δ ) t n , {\displaystyle \quad =\sum _{n=0}^{N}{\frac {(\beta +\delta +1)_{n}(\gamma +1)_{n}}{(\beta +1)_{n}n!}}\operatorname {R} _{n}(\lambda (x);\alpha ,\beta ,\gamma ,\delta )t^{n},} कब α + 1 = − N {\displaystyle \alpha +1=-N\quad } γ + 1 = − N , {\displaystyle \quad \gamma +1=-N,}
2 F 1 ( − x , − x + β − γ ; β + δ + 1 ; t ) 2 F 1 ( x + α + 1 , x + γ + 1 ; α − δ + 1 ; t ) {\displaystyle {}_{2}F_{1}(-x,-x+\beta -\gamma ;\beta +\delta +1;t){}_{2}F_{1}(x+\alpha +1,x+\gamma +1;\alpha -\delta +1;t)} = ∑ n = 0 N ( α + 1 ) n ( γ + 1 ) n ( α − δ + 1 ) n n ! R n ( λ ( x ) ; α , β , γ , δ ) t n , {\displaystyle \quad =\sum _{n=0}^{N}{\frac {(\alpha +1)_{n}(\gamma +1)_{n}}{(\alpha -\delta +1)_{n}n!}}\operatorname {R} _{n}(\lambda (x);\alpha ,\beta ,\gamma ,\delta )t^{n},} कब α + 1 = − N {\displaystyle \alpha +1=-N\quad } β + δ + 1 = − N , {\displaystyle \quad \beta +\delta +1=-N,}
2 F 1 ( − x , − x − δ ; γ + 1 ; t ) 2 F 1 ( x + α + 1 ; x + β + γ + 1 ; α + β − γ + 1 ; t ) {\displaystyle {}_{2}F_{1}(-x,-x-\delta ;\gamma +1;t){}_{2}F_{1}(x+\alpha +1;x+\beta +\gamma +1;\alpha +\beta -\gamma +1;t)} = ∑ n = 0 N ( α + 1 ) n ( β + δ + 1 ) n ( α + β − γ + 1 ) n n ! R n ( λ ( x ) ; α , β , γ , δ ) t n . {\displaystyle \quad =\sum _{n=0}^{N}{\frac {(\alpha +1)_{n}(\beta +\delta +1)_{n}}{(\alpha +\beta -\gamma +1)_{n}n!}}\operatorname {R} _{n}(\lambda (x);\alpha ,\beta ,\gamma ,\delta )t^{n}.} α = a + b − 1 , β = c + d − 1 , γ = a + d − 1 , δ = a − d , x → − a + i x , {\displaystyle \alpha =a+b-1,\beta =c+d-1,\gamma =a+d-1,\delta =a-d,x\rightarrow -a+ix,}
R n ( λ ( − a + i x ) ; a + b − 1 , c + d − 1 , a + d − 1 , a − d ) = W n ( x 2 ; a , b , c , d ) ( a + b ) n ( a + c ) n ( a + d ) n , {\displaystyle \operatorname {R} _{n}(\lambda (-a+ix);a+b-1,c+d-1,a+d-1,a-d)={\frac {\operatorname {W} _{n}(x^{2};a,b,c,d)}{(a+b)_{n}(a+c)_{n}(a+d)_{n}}},} कहाँ W {\displaystyle \operatorname {W} } क्यू-एनालॉग Askey & Wilson (1979) ने मौलिक हाइपरज्यामितीय कार्यों के संदर्भ में परिभाषित क्यू-राकाह बहुपदों को प्रस्तुत किया
p n ( q − x + q x + 1 c d ; a , b , c , d ; q ) = 4 ϕ 3 [ q − n a b q n + 1 q − x q x + 1 c d a q b d q c q ; q ; q ] . {\displaystyle p_{n}(q^{-x}+q^{x+1}cd;a,b,c,d;q)={}_{4}\phi _{3}\left[{\begin{matrix}q^{-n}&abq^{n+1}&q^{-x}&q^{x+1}cd\\aq&bdq&cq\\\end{matrix}};q;q\right].} उन्हें कभी-कभी चर के परिवर्तन के साथ दिया जाता है
W n ( x ; a , b , c , N ; q ) = 4 ϕ 3 [ q − n a b q n + 1 q − x c q x − n a q b c q q − N ; q ; q ] . {\displaystyle W_{n}(x;a,b,c,N;q)={}_{4}\phi _{3}\left[{\begin{matrix}q^{-n}&abq^{n+1}&q^{-x}&cq^{x-n}\\aq&bcq&q^{-N}\\\end{matrix}};q;q\right].}
संदर्भ
↑ Koornwinder, Tom H.; Wong, Roderick S. C.; Koekoek, Roelof; Swarttouw, René F. (2010), "Wilson Class: Definitions" , in Olver, Frank W. J. ; Lozier, Daniel M.; Boisvert, Ronald F.; Clark, Charles W. (eds.), NIST Handbook of Mathematical Functions ISBN 978-0-521-19225-5 MR 2723248 ↑ Koekoek, Roelof; Swarttouw, René F. (1998), The Askey-scheme of hypergeometric orthogonal polynomials and its q-analogue
Askey, Richard; Wilson, James (1979), "A set of orthogonal polynomials that generalize the Racah coefficients or 6-j symbols" (PDF) , SIAM Journal on Mathematical Analysis , 10 (5): 1008–1016, doi :10.1137/0510092 , ISSN 0036-1410 , MR 0541097 , archived from the original on September 25, 2017 Wilson, J. (1978), Hypergeometric series recurrence relations and some new orthogonal functions , Ph.D. thesis, Univ. Wisconsin, Madison
