ANALYSE MATHÉMATIQUE. - Sur une gé néralisation
des polynomes d'Hermite. Note de M.Krawtchouk, transmise par M.
Émile Borel.
M. Krawtchouk. C.R.Acad. Sci. 1929. T.189, No.17. P.620 - 622.
MATHEMATICAL ANALYSIS. - On one generalization of Hermite polynomials.
Note 1 by M. Krawtchouk,
submitted by M. Émile Borel.
.
Let's assume that the polynomial ym(x) of the mth degree
is defined by the following equalities
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�
�
�
�
�
�
�
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u-1
�
i = 0
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piyl(xi)ym(xi) = 0(l � m), = 1 (l = m) |
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�
�
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xi+1-xi = 1, pi � o, |
u-1
�
i = 0
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pi = 1 |
�
�
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. |
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| (1) |
Then we have
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�
�
�
�
�
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xym(x) = m-1ym-1(x)+m0ym(x)+m1ym+1(x) |
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�
�
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mj = |
u-1
�
i = 0
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pixiym(xi)ym+j(xi) |
�
�
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and the minimum of the expression
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Jk(T0,T1,�,Tk-1) = |
u-1
�
i = 0
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pi[T0y0(xi)+�+Tk-1yk-1(xi)-y(xi)]2 (k � u) |
| (2) |
is equal to
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Jk(A0,A1,�,Ak-1) = |
u-1
�
i = 0
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piy2(xi)-A02-�-AN-12 |
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where
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Am = |
u-1
�
i = 0
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piy(xi)ym(xi). |
| (3) |
In the important case p0 = p1 = � = pu-1 which has been studied
by P. Tchebycheff the polynomials ym represent the
generalization of the Legendre polynomials. We want to consider other
important case where
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pi = P(i,u;p,q) = |
�
�
�
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u-1
i
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�
�
�
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piqu-1-i, xi = i (p > 0, q > 0, p+q = 1). |
| (4) |
It is possible to prove that the functions ym have according to
this hypothesis the following simple form:
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�
�
�
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DmP(x-m,u-m;p,q):P(x,u;p,q) |
| (5) | |
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�
�
�
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�
�
�
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u-1
m
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�
�
�
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-1
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(pq)-m |
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m
�
i = o
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(-1)i |
�
�
�
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u-x-1
m-i
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�
�
�
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�
�
�
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x
i
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�
�
�
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pm-iqi |
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concerning the limiting cases of the polynomials
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const. |
x!
ax
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Dm |
�
�
�
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ax-m
(x-m)!
|
�
�
�
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(u� �), p(u-1) = a = const.) |
| (6) |
and the Hermite polynomials
|
const.et2 |
dm
dtm
|
( e-t2) |
�
�
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u� �, x = p(u-1)+t |
| _______
�2pq(u-1)
|
�
�
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. |
| (7) |
In accordance with (4) the formulas (*) and (3) are
conversed respectively to
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| ____________
�(m+1)(u-m-1)pq
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jm+1(x) |
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[ p(u-1)+(q-p)m-x] jm(x)- |
| _______
�m(u-m)pq
|
jm-1(x), |
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�
�
�
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u-1
�
i = 0
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y(xi)DmP(xi-m, u-m; p,q) |
| (8) | |
|
| (-1)m |
�
�
�
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u-1
�
i = 0
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P(i-m,u-m;p,q)Dmy(i-m), |
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Appendices. - 1. Calculation of the incomplete generalized
moments
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Rm(x) = |
x-1
�
i = 0
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P(i,u;p,q)jm(i,u;p,q) (m = 1,2,�,k) |
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of function (4) gives
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Rm(x) = |
�
�
�
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Dm-1P(x-m,u-m;p,q) (m = 1,2,�,k) |
| (9) |
As to kth incomplete factorial moment
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rk(x) = |
x-1
�
i = 0
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P(i,u;p,q) |
�
�
�
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p( u-1) -x
k
|
�
�
�
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(k = 0,1,2,�), |
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it is equal to the linear combination of expressions (9) and moment R0(x) [the result is trivial in a limiting case (7)]. Equality by
M.R. Frisch2
follows from the formula (9) as a special case at (m = 1).
- 2. It is necessary to note the following expansion
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P(x,u;p,q) |
u-1
�
m = 0
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�
�
�
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jm(x,u;p,q) |
| |
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× |
m
�
i = 0
|
(-1)m-i |
�
�
�
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m
i
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�
�
�
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pm-iqi |
�
�
�
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dm[su-m(p1t+q1s)u1]
dtidsm-i
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�
�
�
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s,t = 1
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with the limiting cases which correspond to the polynomials (6) and (7) and are well known.
Footnotes:
1Submitted on September 23, 1929.
2See Ch. Jordan, Statistique math., 1927, p. 85
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