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A p p e a r a n c e
F r o m W i k i p e d i a , t h e f r e e e n c y c l o p e d i a
F
n
(
d
d
x
)
sech
(
x
)
=
sech
(
x
)
P
n
(
tanh
(
x
)
)
.
{\displaystyle F_{n}\left({\frac {d}{dx}}\right)\operatorname {sech} (x )=\operatorname {sech} (x )P_{n}(\tanh(x )).}
where P n is a Legendre polynomial . In terms of generalized hypergeometric functions , they are given by
F
n
(
x
)
=
3
F
2
(
−
n
,
n
+
1
,
1
2
(
x
+
1
)
1
,
1
;
1
)
.
{\displaystyle F_{n}(x )={}_{3}F_{2}\left({\begin{array}{c}-n,~n+1,~{\tfrac {1}{2}}(x+1)\\1,~1\end{array}};1\right).}
Pasternack (1939) generalized the Bateman polynomials to polynomials F m n with
F
n
m
(
d
d
x
)
sech
m
+
1
(
x
)
=
sech
m
+
1
(
x
)
P
n
(
tanh
(
x
)
)
{\displaystyle F_{n}^{m}\left({\frac {d}{dx}}\right)\operatorname {sech} ^{m+1}(x )=\operatorname {sech} ^{m+1}(x )P_{n}(\tanh(x ))}
These generalized polynomials also have a representation in terms of generalized hypergeometric functions, namely
F
n
m
(
x
)
=
3
F
2
(
−
n
,
n
+
1
,
1
2
(
x
+
m
+
1
)
1
,
m
+
1
;
1
)
.
{\displaystyle F_{n}^{m}(x )={}_{3}F_{2}\left({\begin{array}{c}-n,~n+1,~{\tfrac {1}{2}}(x+m+1)\\1,~m+1\end{array}};1\right).}
Carlitz (1957) showed that the polynomials Q n studied by Touchard (1956) , see Touchard polynomials , are the same as Bateman polynomials up to a change of variable: more precisely
Q
n
(
x
)
=
(
−
1
)
n
2
n
n
!
(
2
n
n
)
−
1
F
n
(
2
x
+
1
)
{\displaystyle Q_{n}(x )=(-1)^{n}2^{n}n!{\binom {2n}{n}}^{-1}F_{n}(2x+1)}
Bateman and Pasternack's polynomials are special cases of the symmetric continuous Hahn polynomials .
Examples [ edit ]
The polynomials of small n read
F
0
(
x
)
=
1
{\displaystyle F_{0}(x )=1}
;
F
1
(
x
)
=
−
x
{\displaystyle F_{1}(x )=-x}
;
F
2
(
x
)
=
1
4
+
3
4
x
2
{\displaystyle F_{2}(x )={\frac {1}{4}}+{\frac {3}{4}}x^{2}}
;
F
3
(
x
)
=
−
7
12
x
−
5
12
x
3
{\displaystyle F_{3}(x )=-{\frac {7}{12}}x-{\frac {5}{12}}x^{3}}
;
F
4
(
x
)
=
9
64
+
65
96
x
2
+
35
192
x
4
{\displaystyle F_{4}(x )={\frac {9}{64}}+{\frac {65}{96}}x^{2}+{\frac {35}{192}}x^{4}}
;
F
5
(
x
)
=
−
407
960
x
−
49
96
x
3
−
21
320
x
5
{\displaystyle F_{5}(x )=-{\frac {407}{960}}x-{\frac {49}{96}}x^{3}-{\frac {21}{320}}x^{5}}
;
Properties [ edit ]
Orthogonality [ edit ]
The Bateman polynomials satisfy the orthogonality relation[1] [2]
∫
−
∞
∞
F
m
(
i
x
)
F
n
(
i
x
)
sech
2
(
π
x
2
)
d
x
=
4
(
−
1
)
n
π
(
2
n
+
1
)
δ
m
n
.
{\displaystyle \int _{-\infty }^{\infty }F_{m}(ix )F_{n}(ix )\operatorname {sech} ^{2}\left({\frac {\pi x}{2}}\right)\,dx={\frac {4(-1)^{n}}{\pi (2n+1)}}\delta _{mn}.}
The factor
(
−
1
)
n
{\displaystyle (-1)^{n}}
occurs on the right-hand side of this equation because the Bateman polynomials as defined here must be scaled by a factor
i
n
{\displaystyle i^{n}}
to make them remain real-valued for imaginary argument. The orthogonality relation is simpler when expressed in terms of a modified set of polynomials defined by
B
n
(
x
)
=
i
n
F
n
(
i
x
)
{\displaystyle B_{n}(x )=i^{n}F_{n}(ix )}
, for which it becomes
∫
−
∞
∞
B
m
(
x
)
B
n
(
x
)
sech
2
(
π
x
2
)
d
x
=
4
π
(
2
n
+
1
)
δ
m
n
.
{\displaystyle \int _{-\infty }^{\infty }B_{m}(x )B_{n}(x )\operatorname {sech} ^{2}\left({\frac {\pi x}{2}}\right)\,dx={\frac {4}{\pi (2n+1)}}\delta _{mn}.}
Recurrence relation [ edit ]
The sequence of Bateman polynomials satisfies the recurrence relation[3]
(
n
+
1
)
2
F
n
+
1
(
z
)
=
−
(
2
n
+
1
)
z
F
n
(
z
)
+
n
2
F
n
−
1
(
z
)
.
{\displaystyle (n+1)^{2}F_{n+1}(z )=-(2n+1)zF_{n}(z )+n^{2}F_{n-1}(z ).}
Generating function [ edit ]
The Bateman polynomials also have the generating function
∑
n
=
0
∞
t
n
F
n
(
z
)
=
(
1
−
t
)
z
2
F
1
(
1
+
z
2
,
1
+
z
2
;
1
;
t
2
)
,
{\displaystyle \sum _{n=0}^{\infty }t^{n}F_{n}(z )=(1-t)^{z}\,_{2}F_{1}\left({\frac {1+z}{2}},{\frac {1+z}{2}};1;t^{2}\right),}
which is sometimes used to define them.[4]
References [ edit ]
^ Bateman (1933), p. 28.
^ Bateman (1933), p. 23.
Al-Salam, Nadhla A. (1967). "A class of hypergeometric polynomials" . Ann. Mat. Pura Appl . 75 (1 ): 95–120. doi :10.1007/BF02416800 .
Bateman, H. (1933), "Some properties of a certain set of polynomials." , Tôhoku Mathematical Journal , 37 : 23–38, JFM 59.0364.02
Carlitz, Leonard (1957), "Some polynomials of Touchard connected with the Bernoulli numbers", Canadian Journal of Mathematics , 9 : 188–190, doi :10.4153/CJM-1957-021-9 , ISSN 0008-414X , MR 0085361
Koelink, H. T. (1996), "On Jacobi and continuous Hahn polynomials", Proceedings of the American Mathematical Society , 124 (3 ): 887–898, arXiv :math/9409230 , doi :10.1090/S0002-9939-96-03190-5 , ISSN 0002-9939 , MR 1307541
Pasternack, Simon (1939), "A generalization of the polynomial Fn (x )", London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science , 28 (187): 209–226, doi :10.1080/14786443908521175 , MR 0000698
Touchard, Jacques (1956), "Nombres exponentiels et nombres de Bernoulli", Canadian Journal of Mathematics , 8 : 305–320, doi :10.4153/cjm-1956-034-1 , ISSN 0008-414X , MR 0079021
R e t r i e v e d f r o m " https://en.wikipedia.org/w/index.php?title=Bateman_polynomials&oldid=1193578730 "
C a t e g o r y :
● O r t h o g o n a l p o l y n o m i a l s
● T h i s p a g e w a s l a s t e d i t e d o n 4 J a n u a r y 2 0 2 4 , a t 1 4 : 5 2 ( U T C ) .
● T e x t i s a v a i l a b l e u n d e r t h e C r e a t i v e C o m m o n s A t t r i b u t i o n - S h a r e A l i k e L i c e n s e 4 . 0 ;
a d d i t i o n a l t e r m s m a y a p p l y . B y u s i n g t h i s s i t e , y o u a g r e e t o t h e T e r m s o f U s e a n d P r i v a c y P o l i c y . W i k i p e d i a ® i s a r e g i s t e r e d t r a d e m a r k o f t h e W i k i m e d i a F o u n d a t i o n , I n c . , a n o n - p r o f i t o r g a n i z a t i o n .
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