Download some equations of mixed differences occurring in the theory of

SOME EQUATIONS OF MIXED DIFFERENCES OCCURRING
IN THE THEORY OF PROBABILITY AND THE RELATED
EXPANSIONS IN SERIES OF BESSEL'S FUNCTIONS
B Y H.
BATEMAN.
Equations of mixed differences appear to have been first obtained in the study
of vibrating systems. When John Bernoulli* discussed the problem of a vibrating
string he used a difference equation which is practically equivalent to
dhi
-~ = k2 [un^ 4- un+i - 2un]
(1),
and this difference equation was used by dAlembert to obtain the well-known partial
differential equation
d2u__ Jhi
W ~a~dx2'
Equation (1) plays a prominent part in Euler's investigation of the propagation
of sound in air j* and has been discussed more recently by other writers^:.
The equation is usually solved by forming periodic solutions of the type
un=f(n)eipt,
but I find that for some purposes the particular solution
u
n\l)
~ J2{n—m) (Awt)
can be used more advantageously.
conditions
For instance the solution which satisfies the
u
n (t) =f(n)>
—jT = 0
(ft an integer)
when t = 0, is given generally by the formula^
CO
un(t)=
S J2(n-m) (2kt)f(m).
m= - c o
* Petrop. Cornili. 3 (1728) [32], p. 13. Collected Works, Vol. in. p. 198. The problem of the string of
beads was discussed in detail by Lagrange, Mécanique Analytique, t. i. p. 390.
f Nova theoria lue is et colorum, § 28, Berol. (1746), p. 184. An equation slightly different from the
above has been obtained by Airy in a theory of the aether.
X See for instance M. Born und Th. v. Karman, " Über Schwingungen in Raumgittern," Phys. Zeitschr.
April 15 (1912), p. 297.
§ An expansion of this kind usually converges within a circle \t\ = p ,but with suitable values of the
coefficients f(m) it may converge for all values of t.
19—2
292
H. BATEMAN
Many interesting expansions may be obtained from this result by using particular
solutions un(t), for instance we find that
oo
n2 4- k2t2
m 2 J"2(n_m) (2kt) :
2
m— - c o
this equation holds for integral values of n and for all values of t.
A solution satisfying the conditions
un (t) = 0,
—jj = fc[g(n)— g (n 4- 1)]
(n an integer)
when t = 0 is given generally by the formula
00
Un (t)=
2
m=
g (m) J2{n-m)+i (2fet),
-GO
and this result may also be used to obtain interesting expansions.
Equations of mixed differences have been solved hitherto* either by forming
periodic solutions of the form fp(n)eipt or by using symbolical methods f ; in the case
of the numerous equations^ which occur in the theory of vibrations the method
of periodic solutions is used almost invariably. It will be of interest then to develop
the present method and mention a few problems of chance for wThich the solutions
obtained are naturally adapted.
The equation
^
= ^[Fn^(x)
+ Fn+1(x) ^2Fn(x)]
(2),
which is analogous to the equation of the conduction of heat, possesses the particular
solution e~x In (x) where In (x) denotes the Bessel's function
pJn(ia>).
This particular solution may be generalised so as to provide us with a formula
CO
Fn(x) = e-<*-v 1
I^œ-a)Fm(o)
(3),
which gives the value of Fn(x) when the value of Fm(a) is known§.
* See however the remark on p. 387 of Nielsen's Cylinderfunktionen (1904).
t See for instance J. F. W. Herschel, Calculus of Finite Differences, Cambridge (1820), pp. 37—43;
D. F. Gregory, Mathematical Writings (1865), pp. 38—41; Boole's Finite Differences (1860), pp. 193—207;
G-. Oltramare, Assoc. Franc. Bordeaux (1895), pp. 175—186 ; Calcul de generalisation^ Paris (1899).
X Equations of mixed differences sometimes occur in the theory of radiation, see for instance
Lord Rayleigh's paper " On the propagation of waves along connected systems of similar bodies," Phil.
Mag. XLiv. pp. 356—362 (1897). Scientific Payers, Vol. iv.
§ The solution of the equation
dui7
.
,
which satisfies the conditions un (0) = / (w) is given by
CO
un(x)=
2
Jn-m{x)f{m).
m = - <x>
This formula is due to Sonin, Math. Ann. Bd. xvi. p. 4 (1880).
Thus the particular solution n2 - 2nx + x2 gives rise to the expansion
n2 - 2nx + x2=
GO
S
m— - o o
m2Jn_m (ss).
SOME EQUATIONS OF MIXED DIFFERENCES
293
By using particular solutions we may obtain the following expansions, most
of which are well known:
GO
In<»=
2
tn=
e~xIn(x),
In_m (x - a) Im (a),
Fn(x) =
(m2 + a)In_m(x-a),
Fn(œ) = n* + œ,
eaeos2a cos 2maIn__m (x - a),
Fn(x) = e~^sìn2a cos 2na.
-GO
CO
(n2 + x)ex~a = 2
m — - GO
OD
excos2acos
2na = 2
Mi =
-
GO
The expansion (3) gives the solution of the following problem.
Consider a large number of boxes and a continually increasing large number of
objects each of which is marked with either 4-1 or — 1. The objects are placed in
the boxes, there being no restriction as to the number in each, and when the average
number of objects in a box is a the sum of the numbers in each box is supposed to be
known. The problem is to find the chance that a box chosen at random contains
numbers which add up to n when the average number of objects in each box has
increased to x. It is supposed that an additional object is just as likely to go into
one box as another.
Denoting by Fn (x) the chance that a box contains numbers adding up to n and
treating tc the number of boxes as very large we obtain the difference equation
Fn [x + *) .= ^ [F^ (x) 4- Fn+l (x)] + ( l - ^) Fn (*)
which reduces to (2) when, squares of - are neglected*.
Since
Ir(x-a)^ex-a
2
r— - co
it follows that when Fn (x) is given by (3) we generally have
CO
CO
1
Fn(œ)=
n—-~ co
2
Fn(a).
n— — oo
GO
Consequently if
2
Fn(a) — \, as should be the case, we also have
n— — co
I
? l =
Fn(œ) = l.
- G O
The function e~[x~a) ln_m(x — a) is a type of Green's function for the equation
(2), for a solution of the equation
dun
• - £ (un+14- ««_! - 2un) =f(n, x)
is given by
GO
un =
2
= — CO J
rx
I
e-
(»-a) j n _ m (# _ a \ y ( m> a) da,
— CO
m = — oo J — co
and this solution generally satisfies the conditions
un — 0 for x = — co , or for n — ± oo .
* It is interesting to compare the present problem with those considered by Lord Rayleigh, Theory of
Sound, Vol. i. p. 37. " Dynamical problems in illustration of the kinetic theory of gases," Phil. Mag.
Vol. xxxii. 1891. Scientific Papers, Vol. in.
294
H. BATEMAN
If in our probability problem objects with positive signs only are added after the
initial distribution has been examined the difference equation (2) must be replaced
by the simpler one
dFn
*,'=*•"-*••
<*>•
Poissons formula*
gives the required probability in the case when there is no initial distribution of
objects; from this particular solution we may derive the more general solution
Fn(x)= 2 (x-*y*.<f>(n-p)e-^
(5)
which satisfies the condition Fn(a) = <f>(n). If </>'(w) = 0 when n is negative the
summation extends from p = 0 to p = n and we obtain the solution of the problem
for the case in which all the objects have the positive sign.
Interesting expansions may be derived from (5) by substituting particular
solutions for Fn(x), but the simplest ones are so well known that it will not be
necessary to write them down here.
It is easily seen that in general
co
2
oo
Fn(x) =
1
n-
Fn(a) = l.
-co
It should be mentioned that the problems considered here can be solved
accurately as well as approximately by using partial difference equations in place
of the equations of mixed differences. The equation which takes the place of (4) is
K[Fn(m^l)-Fn(m)-\
= Fn^(m)~Fn(m)
and has already been considered by Lagrange f.
r, ,
N
(6)
A particular solution is given by
m ( m - l ) ... ( m - w 4-1) 1 /-
lyn-n
the well-known formula for the chance that a box contains a number n when we have
no previous knowledge of the distribution of objects among the boxes J. A more
general solution is
Fn(m)=
2
^
Js
r
K-s
1--
Fn_s(r).
If there are no negative signs on the objects we must sum from 0 to n.
the particular solution (7) the formula gives Vandermonde's theorem.
Using
* Recherches sur la probabilité des jugements, Paris (1837), pp. 205—207. See also L. von Bortkewitsch,
Das Gesetz der kleinen Zahlen, Leipzig (Teubner) (1898), this memoir contains tables of the function in
the formula; L. Seidel, Münch. Ber. (1876), pp. 44—50. Smoluchowski, Boltzmann Festschrift (1904).
Bateman, Phil. Mag. (1910) and (1911).
t Miscellanea Taurinensia, t. i. p. 33.
X And the signs are supposed to be all positive.