A modified spherical Bessel function of the first kind (Abramowitz and Stegun 1972), also called a "spherical modified Bessel function of the first kind" (Arfken 1985), is the first solution to the modified spherical Bessel differential equation, given by
 |
(1)
|
where 
is a modified
Bessel function of the first kind (Arfken 1985, p. 633).
For positive 
, the first few values for small nonnegative integer
indices are
 |  |  |
(2)
|
 |  |  |
(3)
|
 |  |  |
(4)
|
 |  |  |
(5)
|
 |  |  |
(6)
|
Writing
 |
(7)
|
the 
are given by the recurrence
equation
 |
(8)
|
together with
 |  |  |
(9)
|
 |  |  |
(10)
|
(Abramowitz and Stegun 1972, p. 443).
For nonnegative integers 
, the analytic continuation
of 
has parity 
(Arfken 1985, p. 633).
is related to the spherical
Bessel function of the first kind 
by
 |
(11)
|
for 
and nonnegative integer 
(Arfken 1985, p. 633).
For 
and nonnegative integers 
, they also satisfy the differential identities
 |  |  |
(12)
|
 |  |  |
(13)
|
and, for 
, the recurrence relations
 |  |  |
(14)
|
 |  |  |
(15)
|
(Arfken 1985, p. 634).
