Solving Differential Equations in Terms of Bessel Functions

14 CHAPTER 1. PRELIMINARIES
Definition 1.15 A generalized hypergeometric series pFq is defined by
α1,α2,...αp
pFq
β1,β2,...βq x
(α1)k · (α2)k ···(αp)k
:=
(β1)k · (β2)k ···(βq)kk! xk , (1.7)
∞
∑
k=0
where (λ)k denotes the Pochhammer symbol
(λ)k := λ · (λ + 1)···(λ + k − 1).
The function is also denoted as pFq α1,α2,...αp;β1,β2,...βq;x .
Example 1.16
Many special functions can be written as a generalized hypergeometric series.
Some well-known series are the exponential and trigonometric series
exp(x) =
∞ x
∑
k=0
k
k! = 0F0(x),
cos(x) = 1 − x2 x4
+
2! 4! − ··· = 0F1
sin(x) = x − x3 x5
+ − ··· = x0F1
3! 5!
−12
−32
−x
2
,
4
−x
2
4
Furthermore, if one of the upper parameters is a negative integer, the series breaks
into a polynomial. But we won’t consider that case.
Theorem 1.17 The generalized hypergeometric series pFq defined in (1.7) satisfies
the differential equation
δ(δ + β1 − 1)···(δ + βq + 1)y(x) = x(δ + α1)···(δ + αp)y(x) (1.8)
where δ = x d dx .
Proof. This can easily be seen if we plug the series pFq into (1.8) and equate
coefficients.
Remarks 1.18
1. For p ≤ q the series pFq is convergent for all z. For p > q + 1 the radius of
convergence is zero, and for p = q + 1 the series converges for |z| < 1.
2. For p ≤ q + 1 the series and its analytic continuation is called a hypergeometric
function.
3. There are identities connecting several hypergeometric functions. A lot
of these formulas can be found in [19] but they can contain typing errors. An
algorithmic approach to check these identities is presented in [17].