An Expansion for the Product of Two Bessel Functions.-1.1. An expansion for the product of two Bessel functions obtained by one of us(1) led to the discovery of a different expansion for the said product multiplied by the leading terms in the power series for the Bessel functions. Two proofs of this second expansion are given here

Bateman, H.

Rice, S. O.

English

Caltech Authors - Main

PHYSICS: BA TEMAN AND RICE
di'
-
Poxi ~~~~~(5.4)
The hyperplane at infinity for these affine coordinates dx' is the inter-
section of p,oZ' = 0 with ZW = 0. The result of combming (5.3) with
(5.4) is
P-i- z-oidxi = (5.5)
and these equations are inverse to (5.2).
10. Veblen, Projcktive Relativitstheorie, Berlin, 1933.
21This matrix is, of course, a modification of the one introduced by T. Y. Thomas in
these PROCEEDINGS, 12, 356 (1926).
' This pseudo group may be discussed in much the same way as what I called the
enlarged conformal group in these PROCEEDINGS, 14, 738 (1928).
4J. L. Vanderslice, these PROCEEDINGS, 20, 674 (1934).
' These differ slightly from the formulas of T. Y. Thomas (see reference in Note 2)
because of the difference in our matrices of transformation and because the second co-
variant index of our connection runs from 1 to n instead of from 0 to n as with Thomas.
I am indebted to Dr. Vanderslice for making the computation.
' The references are given in Vanderslice's paper (Note 4 above).
SOME EXPANSIONS ASSOCIATED WITH BESSEL FUNCTIONS
By H. BATEMAN AND S. 0. RICE
NoRmAN BRIDGE LABORATORY OF PHYSICS AND BELL TELEPHONE LABORATORIBS
Communicated January 24, 1935
1. An ExPansion for the Product of Two Bessd Functions.-1.1. An ex-
pansion for the product of two Bessel functions obtained by one of us'
led to the discovery of a different expansion for the said product multiplied
by the leading terms in the power series for the Bessel functions. Two
proofs of this second expansion are given here. The first depends upon the
fact that the function V = Ja(u)uaJ,(v)VC satisfies the equation
61V a1V 12V 4X a + (1
-2a) a-+ uV + v + (1 -2c) - + vV= 0
which is transformed by the substitutions u = sx, v = z(l - x) into
Z2 a2v+(_ )zv+ [Z2-(n + 1)(2r + 2n -1)]+Ja2 3- 2r)s- (2r- -+1(2n+21) IV
+x(i -x) j12+ [2s+2+ (2r - 3)x1b-+ (2n+1)(2r7+2nl-1)V=O
VOL. 21, 1935 173
PHYSICS: BATEMAN AND RICE
where r = a + c + 1/2, s = -a -1/ and n is an arbitrary constant. A
particular solution of the last equation is
V = =VX7 Jr +2n (z) D2X [A+2f (1 _)2c+h].
If we assume that the former solution can be expressed as the sum of terms
of type A.V, where n runs from 0 to co, the constant A. may be found by
letting x approach zero and comparing our assumed series with Sonine's
expansion2
(r + 2n) r(r + n)
ZSJr+s(z) =2-s Z; GD) r2x
n-O II (r + s + n) J+2()
It is thus found that
Ja(u)uVJc(v)VC = 2 AxVsis-O
where
Ax_ (_)n (r + 2n) I[ (r + n - 1)22'x f (a + n) II(c + n)n!
1.2. An independent proof of the validity of the expansion is based upon
the equation'
J8(zx)Jc(Z - zx) = xa(1 -_x)CA/J -a(zW)w-a-Cta-a (1 - )-C dt
where
47rA = i + cot 7r(a +c), w2 =-t + X)
and the contour P is the double loop (1+, 0+, 1-, 0-) of the Pochham-
mer type. The arguments of t and (1 -t) are zero at the starting point
which lies on the real axis between the points t = 0 and t = 1. By using
the expansion2
WG4-CJa+c(zw) = (z/2)a+c-bz II(b+n-1)(b+2n)
u=O n!, II(a + c)
F(b + n, -n; a + c + 1; w2)Jb+2X(z) (1)
which is valid when b and a + c are not negative integers, and integrating
termwise we obtain terms involving integrals of type
I = A JF(b + n, -n; a + c + 1; W2)1al(1-t)-c `dt.
p
1-74 PROC. N. A. S.
PHYSICS: BA TEMAN AND RICE
The series may be shown to converge uniformly with respect to t when b is
not an integer and so the integration of the series is permissible. Indeed,
when n is a positive integer,
r(b + n)F(b + n, -n; a + c + 1; w2)/[ll(a + c)r(b + n - a - c)] = A'
(T - w2)dTr/[Ta+c+n+1(1 - ,)b+n-a-cI (2)
where A' = i + cot irb, as may be verified by expanding ('r- w2)¶. This
relation defines the hypergeometric function for all values of w. Choosing
the loops of the contour P to be circles of unit diameter with centers at
0, 1, respectively, we see that the total length of P is 4wr and that on P
we have the inequalities
1 r w2 < + I w2 | T -a-c-n-l 1 2"+'Bi, 1(1 _ r)a+c-b-xI <2nB2
where logB, = IR(a + c)| log 2 + 27r|I(a + c)|, log B2 = jR(b - a - c)1 log
2 + 2w11(b-a - c). Thus the modulus of the right-hand side of (2) is
less than
i + cot (7rb) 3 + I W2 |) 2+BjB2
and so the modulus of the nth term in (1) is less than
(b+2n)r(b+n -a-c) (i+ cot(wb)) - + w21) 22X+lBlB2Jb+2x(z)/n!
As n-> c, r(b + 2n + 1)Jb+2n(z) > (z/2)b+'" and sotheseriesconverges
uniformly for all finite values of w2 unless b is an integer. When b is an
integer further investigation is needed to establish the uniform convergence
but this may be avoided by excluding this case for the present and then re-
turning later to an examination of the final result by the method of con-
tinuity.
1.3. The integral I may be expressed as a hypergeometric function of x
when b = a + c + 1/2. In order to carry through the proof we assume for
the present that neither a + c + 1/2 nor a + c is an integer. We first use
K:ummer's relation
F(p, q; p + q + 1/2; 4y - 4y2) = F(2p, 2q; p + q + 1/2; Y)
wherein p = a + c + n + 1/2, q = -n, 4y(1-y) = w2, 2y = 1 + i
(t - X)-t-1'2 (1 - t)-1/2. The usual restrictions 12yj < 1, 14y(1 -y)I < 1
are not necessary here since the series terminate and each side is merely a
VOL. 21, 1935 175
PHYSICS: BATEMANAND RICE
polynomial in y. The resulting series is then arranged in ascending powers
of x and the hypergeometric function in the integrand becomes
2n (2a + 2c + 2n + 1), (-2n), ( -ix r2 I Ar(t)
r MO r!(a + c + 1)r k2N/t(1i-t
where
1
A,(t) = F(2a+2c+2n+r+ I,r-2n;a+c+r+ 1; + -0
= (-)#iT[t/(1 -t) 1 1/2rFQ r-n,4 r + 2--n; a + c + r + 1; iii).
The last step may be verified by using Kummer's formula combined with
the relation between hypergeometric functions of arguments y and y/
(y - 1). It is convenient to perform the transformations at the point
t = Nexp iwr, arg (1 - t) = 0, where N is a large positive constant, and then
extend the result to the contour P by analvtical continuation. When
written in full the expression for I is
f2n m (2a+2c+2n+1)r (-2n), ('/2r-n) ('/2r+'/2-n)Ar
JP r Os-Or!(a+c+l)r2rs!(a+c+r+1)sta+ + (1-
wherem < (n - '/2r). Using now Pochhammer's extension of the Eulerian
integral of the first kind and summing the resultant hypergeometric series
in s of argument unity by means of the formula of Gauss, we have finally
]l[(a + c)fl(a + n . '/2) 2n (2a + 2c + 2n + 1)r (-2n)rX
ll(c + n) r-o2r'I(a + 1/2r) II(a + 1/2r-1/2) r!
1IH(a + c)J1I(2a + 2n)
.2xn (a+n)fl(c + 2a) F(2a + 2c + 2n + 1, -2n; 2a + 1; x).
When this hypergeometric function is expressed as a Jacobi polynomial and
the corresponding value of I is placed in the infinite series for Ja(zx)
J,(z - zx) we obtain the expansion given in section 1.1
co (a+c+2n+1/2) ll(a+c+n-'/,)Xa(1 - x)cJ.(zx) J'(z - zx) = 2 (-)fl 2 I a+n)11c+nn(1) )c() #o () 22n1/2 II(a + n) D(c + n)n!
Z1/2Ja+c+2n+2/x(Z)D2[x _2a+2n(l _ X)2c+2n].
The restriction that a + c be not an integer and that a + c + '/2 be not a
positive integer may be removed by the use of continuity but the restriction
that a + c + 1/2 be not a negative integer is necessary. Using Lommel's
expansion to compare the expansion for Ja(zx) Jc(z - zx), mentioned in
176 PROC. N. A. S.
PHYSICS: BA TEMANAND RICE
the introduction, with the expansion developed here we obtain the equa-
tion
k
F(-2k, a + c + 1; a-k + 1;x)= CnF(-2n, 2a + 2c + 2n + 1;
n O
2a+ 1; x)
where
(-)u+k(a+c+2n+142)H(2a+2c+2n)ll(a+n1-/2)fl(a-k)fl(c+k) (2k) !
C" fik+2dll(a + c + k + n + 1/,)II(c + n)1I(2a)Il(a + c)(2n)! (k-n)l
2. Generating Functions for Bessel's Functions of Non-Integral Order.
If R(P) > 0,
eX7 exp ((y - yt)/I)t'' dt = e XT E )r'T/(v)r+=
JO0 r=O
co co
E E (_X)yTsyW1[S!(v)r+lI.
s-0 r=O
Since the double series is absolutely convergent we may write it in t,he form
E b)/.r), E (-_tY),1[S!(V).+s'+l]+ (_Xr), F, (_-,y)r/[(P),+,(n+r)!]
n-0 s-0 n=l r=O
where in the first series n = r - s and in the second series n = s - r, con-
sequently when R(v) > 0
r(P)JO,exp ((y-yt)/r) tV' di = E (y/T)l(xyy)-n/2-/2 J,+,(2Vx) +
- I U-0
E (-xSr)"F(1; n + 1, v + 1; -xy)/[n!r(v + 1)].
X1
If we multiply both sides of this equation by exp (-ux) xm -1, where
R(u), R(m), R(u + T) > 0, and integrate between x = 0 and x = o we
obtain a convergent integral on the left. Moreover, when jrj < Jul the
series on the right may be integrated term by term and the factor (u + T)m
arising from integration on the left may be expanded by the binomial theo-
rem. Upon arranging the left-hand side as a Laurent series in r and com-
paring it with the expression on the right we find
1+P
y17(m)+ ( + l)F(m; n+v+1; -y/u)=, xm- 2 -IJ5+ (2Vxy)dx,
umr(n + + 1)
IX7 (u) fexuxm+'-1F(l; n + 1, v + 1; -xy)dx
VOL. 21, 1935 177
PHYSICS: BATEMANAND RICE
= irm ^r(m + n)F(l, mt + n; n + 1,v+ 1; -y/u).
(3)
The first integral is well known and the second may be readily verified
when R(u) > 0, by expanding the hypergeometric function, and integrating
termwise.
When u = 0 the integral 1:,, converges only if 0 < m + n < 1, as may
be seen by Barnes' asymptotic formula for 1F2, assuming y real and posi-
tive. If, in addition, m + n -v is a positive integer it may be shown that
the integral is zero by letting u - 0 in the hypergeometric expression for
Im,. The general value of Hardy's type5 thus obtained is also the actual
value since the integral converges. To do this it is convenient to use the
reduction formula
(m + n-1)yF(1, m + n; n + 1, v + 1; -y/u)
= tu [1 - F(1, m- + n - 1; n,v; -y/u)]
to obtain
yIXM,V = InvuI-m-xr(m + n-lFl 1n, 1 -v; 2 -m-n; u/y) +
3Xu-Mn!r(m)(n- ' - 1)xF(m; v- n + 1; - y/u) (4)
This is merely the result of a formal transformation of the function 2F2
and holds without restriction. It is, in fact, a particular case of Barnes'
asymptotic formula4 for F(a, c; u, v; z) which holds when arg zj <3T/2.
Making use of Kummer's relation
F(m; '- n + 1; -y/u) = exp (-y/u)F(v - n - m + 1; v- n + 1; y/u)
we note that the confluent hypergeometric function on the right is a poly-
nomial when n + m - v is a positive integer. The last term in (4) then
becomes zero as u -"0. If m + n < 1 the first term in (4) also tends
zero and so in these circumstances the actual value of 1I, is zero.
This result may be compared with Orr's formula5
x r(c7) r'(a, -m)
_tx F(ai, a2, ..., a.; cl, c2, . . ., c,;-x)dx = r(m) III
(m > a, where at is the algebraical least a,) and with the result of his re-
mark that by putting x= z/a, and making a,-o o, a corresponding formula
may be derived for the case in which the number of the a's is less than the
number of c's.
Several definite integrals which partake of the nature of generating
functions are
fhJ (z dn u)sn21 ucn2P-1ludn1+`udu=E k?x(_)() B(v+fwp)J ,,(z)
178 PR.OC. N. A. S.
CORRECTION: R. W. WILSON
K m
~~~~~~~~2x(Z5B(p + n,p)j J,,(z dn u)sn2-lu cn2plu dn'-u du=-> !22 j)+n (z)
JJ,(zx)xP+l exp[1/2tz(l - x2)]dx = E t" z` Jh+M+l (z)
n=0
11
IJ (zx)xe+l exp [1/2t(1 -x2)] dx = E t" z-'-' J +x÷i (z).
,./ n~~~~~~~-O
In the first two expressions the Bessel functions of the first kind (J,'s)
may be replaced throughout by functions of the second kind (Y,'s); and
further, if the factor (-)" in the summation involving J,,-x(z) be omitted
the functions of the first kind may be replaced throughout by modified
functions of the second kind (K,'s). These equations, which hold for
R(v), R(p) > 0, may be verified by replacing z dn u by VZ2 -z2k2skn2u, ex-
panding the Bessel functions in the integrand, using Lommel's expansion,
and integrating termwise.
The last two equations, which hold when R(js) > -1 and which may be
verified by using"
zn+1 1l
J^+n+)= fX!J (ZX)XI+l (1 -X2)xdx,
have probably been noticed before. The analogous series Ftez"Jn+,(z)/n!'
has already been summed by W. Kapteyn7 and some other Dutch mathe-
maticians.
1 S. 0. Rice, "On Contour Integrals for the Product of Two Bessel Functions," Quart.
Jour. Math.
2 G. N. Watson, Theory of Bessel Functions, 15.21.
3Ibid., §5.23.
4E. W. Barnes, Trans. Roy. Soc. Lond., 206, Part XI, 295 (1906).
6 G. H. Hardy, Trans. Camb. Phil. Soc., 21, 13 (1908); W. McF. Orr, Ibid., 17, 171
(1898); 19, 151 (1900).
6 G. N. Watson, 1. c., §12.11.
7 W. Kapteyn, Vraagstuk, 132, Solutions by G. R. Boogaerdt't, S. C. van Veen and
others. Wiskundige Opgaven, 14, 269-271 (1928).
SIMIMYS, A NEW NAME TO REPLACE EUMYSOPS WILSON,
PREOCCUPIED.-A CORRECTION
By ROBERT W. WILSON
A description of a new genus of cricetine-like rodent was published in the
January, 1935, issue of the PROCEEDINGS of the National Academy of
VOL. 21, 1935 179


Some expansions associated with Bessel functions

PHYSICS: BA TEMAN AND RICE
di'
-
Poxi ~~~~~(5.4)
The hyperplane at infinity for these affine coordinates dx' is the inter-
section of p,oZ' = 0 with ZW = 0. The result of combming (5.3) with
(5.4) is
P-i- z-oidxi = (5.5)
and these equations are inverse to (5.2).
10. Veblen, Projcktive Relativitstheorie, Berlin, 1933.
21This matrix is, of course, a modification of the one introduced by T. Y. Thomas in
these PROCEEDINGS, 12, 356 (1926).
' This pseudo group may be discussed in much the same way as what I called the
enlarged conformal group in these PROCEEDINGS, 14, 738 (1928).
4J. L. Vanderslice, these PROCEEDINGS, 20, 674 (1934).
' These differ slightly from the formulas of T. Y. Thomas (see reference in Note 2)
because of the difference in our matrices of transformation and because the second co-
variant index of our connection runs from 1 to n instead of from 0 to n as with Thomas.
I am indebted to Dr. Vanderslice for making the computation.
' The references are given in Vanderslice's paper (Note 4 above).
SOME EXPANSIONS ASSOCIATED WITH BESSEL FUNCTIONS
By H. BATEMAN AND S. 0. RICE
NoRmAN BRIDGE LABORATORY OF PHYSICS AND BELL TELEPHONE LABORATORIBS
Communicated January 24, 1935
1. An ExPansion for the Product of Two Bessd Functions.-1.1. An ex-
pansion for the product of two Bessel functions obtained by one of us'
led to the discovery of a different expansion for the said product multiplied
by the leading terms in the power series for the Bessel functions. Two
proofs of this second expansion are given here. The first depends upon the
fact that the function V = Ja(u)uaJ,(v)VC satisfies the equation
61V a1V 12V 4X a + (1 -2a) a-+ uV + v + (1 -2c) - + vV= 0
which is transformed by the substitutions u = sx, v = z(l - x) into
Z2 a2v+(_ )zv+ [Z2-(n + 1)(2r + 2n -1)]+Ja2 3- 2r)s- (2r- -+1(2n+21) IV
+x(i -x) j12+ [2s+2+ (2r - 3)x1b-+ (2n+1)(2r7+2nl-1)V=O
VOL. 21, 1935 173
PHYSICS: BATEMAN AND RICE
where r = a + c + 1/2, s = -a -1/ and n is an arbitrary constant. A
particular solution of the last equation is
V = =VX7 Jr +2n (z) D2X [A+2f (1 _)2c+h].
If we assume that the former solution can be expressed as the sum of terms
of type A.V, where n runs from 0 to co, the constant A. may be found by
letting x approach zero and comparing our assumed series with Sonine's
expansion2
(r + 2n) r(r + n)
ZSJr+s(z) =2-s Z; GD) r2xn-O II (r + s + n) J+2()
It is thus found that
Ja(u)uVJc(v)VC = 2 AxVs
is-O
where
Ax_ (_)n (r + 2n) I[ (r + n - 1)
22'x f (a + n) II(c + n)n!
1.2. An independent proof of the validity of the expansion is based upon
the equation'
J8(zx)Jc(Z - zx) = xa(1 -_x)CA/J -a(zW)w-a-Cta-a (1 - )-C dt
where
47rA = i + cot 7r(a +c), w2 =-t + X)
and the contour P is the double loop (1+, 0+, 1-, 0-) of the Pochham-
mer type. The arguments of t and (1 -t) are zero at the starting point
which lies on the real axis between the points t = 0 and t = 1. By using
the expansion2
WG4-CJa+c(zw) = (z/2)a+c-bz II(b+n-1)(b+2n)
u=O n!, II(a + c)
F(b + n, -n; a + c + 1; w2)Jb+2X(z) (1)
which is valid when b and a + c are not negative integers, and integrating
termwise we obtain terms involving integrals of type
I = A JF(b + n, -n; a + c + 1; W2)1al(1-t)-c `dt.
p
1-74 PROC. N. A. S.
PHYSICS: BA TEMAN AND RICE
The series may be shown to converge uniformly with respect to t when b is
not an integer and so the integration of the series is permissible. Indeed,
when n is a positive integer,
r(b + n)F(b + n, -n; a + c + 1; w2)/[ll(a + c)r(b + n - a - c)] = A'
(T - w2)dTr/[Ta+c+n+1(1 - ,)b+n-a-cI (2)
where A' = i + cot irb, as may be verified by expanding ('r- w2)¶. This
relation defines the hypergeometric function for all values of w. Choosing
the loops of the contour P to be circles of unit diameter with centers at
0, 1, respectively, we see that the total length of P is 4wr and that on P
we have the inequalities
1 r w2 < + I w2 | T -a-c-n-l 1 2"+'Bi, 1(1 _ r)a+c-b-xI <2nB2
where logB, = IR(a + c)| log 2 + 27r|I(a + c)|, log B2 = jR(b - a - c)1 log
2 + 2w11(b-a - c). Thus the modulus of the right-hand side of (2) is
less than
i + cot (7rb)
3
+ I W2 |) 2+BjB2
and so the modulus of the nth term in (1) is less than
(b+2n)r(b+n -a-c) (i+ cot(wb)) - + w21) 22X+lBlB2Jb+2x(z)/n!
As n-> c, r(b + 2n + 1)Jb+2n(z) > (z/2)b+'" and sotheseriesconverges
uniformly for all finite values of w2 unless b is an integer. When b is an
integer further investigation is needed to establish the uniform convergence
but this may be avoided by excluding this case for the present and then re-
turning later to an examination of the final result by the method of con-
tinuity.
1.3. The integral I may be expressed as a hypergeometric function of x
when b = a + c + 1/2. In order to carry through the proof we assume for
the present that neither a + c + 1/2 nor a + c is an integer. We first use
K:ummer's relation
F(p, q; p + q + 1/2; 4y - 4y2) = F(2p, 2q; p + q + 1/2; Y)
wherein p = a + c + n + 1/2, q = -n, 4y(1-y) = w2, 2y = 1 + i
(t - X)-t-1'2 (1 - t)-1/2. The usual restrictions 12yj < 1, 14y(1 -y)I < 1
are not necessary here since the series terminate and each side is merely a
VOL. 21, 1935 175
PHYSICS: BATEMANAND RICE
polynomial in y. The resulting series is then arranged in ascending powers
of x and the hypergeometric function in the integrand becomes
2n (2a + 2c + 2n + 1), (-2n), ( -ix r2 I Ar(t)
r MO r!(a + c + 1)r k2N/t(1i-t
where
1
A,(t) = F(2a+2c+2n+r+ I,r-2n;a+c+r+ 1; + -0
= (-)#iT[t/(1 -t) 1 1/2rFQ r-n,4 r + 2--n; a + c + r + 1; iii).
The last step may be verified by using Kummer's formula combined with
the relation between hypergeometric functions of arguments y and y/
(y - 1). It is convenient to perform the transformations at the point
t = Nexp iwr, arg (1 - t) = 0, where N is a large positive constant, and then
extend the result to the contour P by analvtical continuation. When
written in full the expression for I is
f2n m (2a+2c+2n+1)r (-2n), ('/2r-n) ('/2r+'/2-n)Ar
JP r Os-Or!(a+c+l)r2rs!(a+c+r+1)sta+ + (1-
wherem < (n - '/2r). Using now Pochhammer's extension of the Eulerian
integral of the first kind and summing the resultant hypergeometric series
in s of argument unity by means of the formula of Gauss, we have finally
]l[(a + c)fl(a + n . '/2) 2n (2a + 2c + 2n + 1)r (-2n)rX
ll(c + n) r-o2r'I(a + 1/2r) II(a + 1/2r-1/2) r!
1IH(a + c)J1I(2a + 2n)
.2xn (a+n)fl(c + 2a) F(2a + 2c + 2n + 1, -2n; 2a + 1; x).
When this hypergeometric function is expressed as a Jacobi polynomial and
the corresponding value of I is placed in the infinite series for Ja(zx)
J,(z - zx) we obtain the expansion given in section 1.1
co (a+c+2n+1/2) ll(a+c+n-'/,)Xa(1 - x)cJ.(zx) J'(z - zx) = 2 (-)fl 2 I a+n)11c+nn(1) )c() #o () 22n1/2 II(a + n) D(c + n)n!
Z1/2Ja+c+2n+2/x(Z)D2[x _2a+2n(l _ X)2c+2n].
The restriction that a + c be not an integer and that a + c + '/2 be not a
positive integer may be removed by the use of continuity but the restriction
that a + c + 1/2 be not a negative integer is necessary. Using Lommel's
expansion to compare the expansion for Ja(zx) Jc(z - zx), mentioned in
176 PROC. N. A. S.
PHYSICS: BA TEMANAND RICE
the introduction, with the expansion developed here we obtain the equa-
tion
k
F(-2k, a + c + 1; a-k + 1;x)= CnF(-2n, 2a + 2c + 2n + 1;
n O
2a+ 1; x)
where
(-)u+k(a+c+2n+142)H(2a+2c+2n)ll(a+n1-/2)fl(a-k)fl(c+k) (2k) !
C" fik+2dll(a + c + k + n + 1/,)II(c + n)1I(2a)Il(a + c)(2n)! (k-n)l
2. Generating Functions for Bessel's Functions of Non-Integral Order.
If R(P) > 0,
eX7 exp ((y - yt)/I)t'' dt = e XT E )r'T/(v)r+=
JO0 r=O
co co
E E (_X)yTsyW1[S!(v)r+lI.
s-0 r=O
Since the double series is absolutely convergent we may write it in t,he form
E b)/.r), E (-_tY),1[S!(V).+s'+l]+ (_Xr), F, (_-,y)r/[(P),+,(n+r)!]
n-0 s-0 n=l r=O
where in the first series n = r - s and in the second series n = s - r, con-
sequently when R(v) > 0
r(P)JO,exp ((y-yt)/r) tV' di = E (y/T)l(xyy)-n/2-/2 J,+,(2Vx) +
- I U-0
E (-xSr)"F(1; n + 1, v + 1; -xy)/[n!r(v + 1)].
X1
If we multiply both sides of this equation by exp (-ux) xm -1, where
R(u), R(m), R(u + T) > 0, and integrate between x = 0 and x = o we
obtain a convergent integral on the left. Moreover, when jrj < Jul the
series on the right may be integrated term by term and the factor (u + T)m
arising from integration on the left may be expanded by the binomial theo-
rem. Upon arranging the left-hand side as a Laurent series in r and com-
paring it with the expression on the right we find
1+P
y17(m)+ ( + l)F(m; n+v+1; -y/u)=, xm- 2 -IJ5+ (2Vxy)dx,
umr(n + + 1)
IX7 (u) fexuxm+'-1F(l; n + 1, v + 1; -xy)dx
VOL. 21, 1935 177
PHYSICS: BATEMANAND RICE
= irm ^r(m + n)F(l, mt + n; n + 1,v+ 1; -y/u).
(3)
The first integral is well known and the second may be readily verified
when R(u) > 0, by expanding the hypergeometric function, and integrating
termwise.
When u = 0 the integral 1:,, converges only if 0 < m + n < 1, as may
be seen by Barnes' asymptotic formula for 1F2, assuming y real and posi-
tive. If, in addition, m + n -v is a positive integer it may be shown that
the integral is zero by letting u - 0 in the hypergeometric expression for
Im,. The general value of Hardy's type5 thus obtained is also the actual
value since the integral converges. To do this it is convenient to use the
reduction formula
(m + n-1)yF(1, m + n; n + 1, v + 1; -y/u)
= tu [1 - F(1, m- + n - 1; n,v; -y/u)]
to obtain
yIXM,V = InvuI-m-xr(m + n-lFl 1n, 1 -v; 2 -m-n; u/y) +
3Xu-Mn!r(m)(n- ' - 1)xF(m; v- n + 1; - y/u) (4)
This is merely the result of a formal transformation of the function 2F2
and holds without restriction. It is, in fact, a particular case of Barnes'
asymptotic formula4 for F(a, c; u, v; z) which holds when arg zj <3T/2.
Making use of Kummer's relation
F(m; '- n + 1; -y/u) = exp (-y/u)F(v - n - m + 1; v- n + 1; y/u)
we note that the confluent hypergeometric function on the right is a poly-
nomial when n + m - v is a positive integer. The last term in (4) then
becomes zero as u -"0. If m + n < 1 the first term in (4) also tends
zero and so in these circumstances the actual value of 1I, is zero.
This result may be compared with Orr's formula5
x r(c7) r'(a, -m)_tx F(ai, a2, ..., a.; cl, c2, . . ., c,;-x)dx = r(m) III
(m > a, where at is the algebraical least a,) and with the result of his re-
mark that by putting x= z/a, and making a,-o o, a corresponding formula
may be derived for the case in which the number of the a's is less than the
number of c's.
Several definite integrals which partake of the nature of generating
functions are
fhJ (z dn u)sn21 ucn2P-1ludn1+`udu=E k?x(_)() B(v+fwp)J ,,(z)
178 PR.OC. N. A. S.
CORRECTION: R. W. WILSON
K m ~~~~~~~~2x(Z5B(p + n,p)
j J,,(z dn u)sn2-lu cn2plu dn'-u du=-> !22 j)+n (z)
JJ,(zx)xP+l exp[1/2tz(l - x2)]dx = E t" z` Jh+M+l (z)
n=0
11
IJ (zx)xe+l exp [1/2t(1 -x2)] dx = E t" z-'-' J +x÷i (z).
,./ n~~~~~~~-O
In the first two expressions the Bessel functions of the first kind (J,'s)
may be replaced throughout by functions of the second kind (Y,'s); and
further, if the factor (-)" in the summation involving J,,-x(z) be omitted
the functions of the first kind may be replaced throughout by modified
functions of the second kind (K,'s). These equations, which hold for
R(v), R(p) > 0, may be verified by replacing z dn u by VZ2 -z2k2skn2u, ex-
panding the Bessel functions in the integrand, using Lommel's expansion,
and integrating termwise.
The last two equations, which hold when R(js) > -1 and which may be
verified by using"
zn+1 1l
J^+n+)= fX!J (ZX)XI+l (1 -X2)xdx,
have probably been noticed before. The analogous series Ftez"Jn+,(z)/n!'
has already been summed by W. Kapteyn7 and some other Dutch mathe-
maticians.
1 S. 0. Rice, "On Contour Integrals for the Product of Two Bessel Functions," Quart.
Jour. Math.
2 G. N. Watson, Theory of Bessel Functions, 15.21.
3Ibid., §5.23.
4E. W. Barnes, Trans. Roy. Soc. Lond., 206, Part XI, 295 (1906).
6 G. H. Hardy, Trans. Camb. Phil. Soc., 21, 13 (1908); W. McF. Orr, Ibid., 17, 171
(1898); 19, 151 (1900).
6 G. N. Watson, 1. c., §12.11.
7 W. Kapteyn, Vraagstuk, 132, Solutions by G. R. Boogaerdt't, S. C. van Veen and
others. Wiskundige Opgaven, 14, 269-271 (1928).
SIMIMYS, A NEW NAME TO REPLACE EUMYSOPS WILSON,
PREOCCUPIED.-A CORRECTION
By ROBERT W. WILSON
A description of a new genus of cricetine-like rodent was published in the
January, 1935, issue of the PROCEEDINGS of the National Academy of
VOL. 21, 1935 179


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Some expansions associated with Bessel functions

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