On*eneralised Hankel)unctions and a%ifurcation of Their$symptotic
Expansion
L. M. B. C. CAMPOS, M. J. S. SILVA
CCTAE, IDMEC, LAETA,
Instituto Superior Técnico, Universidade de Lisboa,
Av. Rovisco Pais 1, 1049-001, Lisboa,
PORTUGAL
Abstract: The generalised Bessel differential equation has an extra parameter relative to the original Bessel equa-
tion and its asymptotic solutions are the generalised Hankel functions of two kinds distinct from the original
Hankel functions. The generalised Bessel differential equation of order νand degree µreduces to the original
Bessel differential equation of order νfor zero degree, µ= 0. In both cases the differential equations have a
regular singularity near the origin and the the point at infinity is the other singularity. The point at infinity is an
irregular singularity of different degree, namely one for the original and two for the generalised Bessel differen-
tial equation. It follows that in the limit of degree being equal to zero the generalised Hankel functions do not
converge to the original ones. The implication is that the generalised Bessel differential equation has a Hopf-type
bifurcation for the asymptotic solution. In the case of a real variable and parameters the asymptotic solution is: (i)
oscillatory when the degree of generalised Hankel function is zero (corresponding in this case to original Hankel
functions); (ii) diverging hence unstable for the generalised Hankel functions with positive degree; (iii) decaying
hence stable for the generalised Hankel functions with negative degree.
Key-Words: Generalised Bessel differential equation, Generalised Bessel functions, Generalised Hankel
functions, Asymptotic solutions, Irregular singularities, Thomé normal integrals
Received: October 21, 2023. Revised: February 22, 2024. Accepted: March 11, 2024. Published: April 10, 2024.
1 Introduction
The Bessel differential equation was first considered
in connection with the oscillations of a heavy chain,
[1], and vibrations of a circular membrane, [2], and,
since the work by [3], has had a vast number of ap-
plications supported by an extensive theory, [4], [5].
The original Bessel differential equation appears in
connection with several problems in mathematical
physics and engineering, notably associated not only
with cylindrical and spherical, [6], [7], [8], [9], ge-
ometries but also with hypercylindrical and hyper-
spherical, [10], geometries, including the propaga-
tion of acoustic, [11], [12], [13], and electromagnetic,
[14], [15], [16], waves, heat diffusion, [17], [18],
[19], quantum mechanics, [20], [21], [22], and also
problems in solid mechanics, [23], [24], [25], such as
oscillations of heavy chains and vibrations of circular
elastic plates, [26], [27], [28].
Among the applications to waves in fluids, [29],
[30], [31], are mentioned two: (i) sound in a cylindri-
cal nozzle with uniform axial flow, corresponding to
longitudinal compressive waves; (ii) vortical waves
in a rotating flow with constant angular velocity, cor-
responding to transverse, incompressible modes. Al-
though in both cases (i) and (ii) the pressure perturba-
tion has a radial dependence specified by the original
cylindrical Bessel differential equation, there are two
differences: (a) the wave speed resulting from the dis-
persion relation between frequency and wavevector;
(b) the polarisation relations relating the pressure per-
turbations to other wave variables such as velocity,
density and temperature perturbations for adiabatic
propagation, since entropy modes are excluded. The
coupling of (i) and (ii) for acoustic-vortical waves in a
compressible swirling flow with uniform axial veloc-
ity and constant angular velocity no longer satisfies
the original Bessel differential equation, but rather
the generalised Bessel differential equation consid-
ered in the present paper, due to the physical interac-
tion between compressibility and rigid body rotation:
(α) for small radius the swirl velocity is small and
the acoustic-vortical waves resemble acoustic waves,
leading to the usual regular singularity on axis; (β)
when the radius becomes larger the swirl velocity also
increases, consequently the irregular singularity of
degree one at infinity for purely acoustic or vortical
waves becomes of degree two due to coupling of ro-
tation and compressibility in acoustic-vortical shear
waves.
Since the singularities of the differential equation
affect the solution everywhere, in the case of super-
position of a uniform mean flow and a rigid body
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rotation, the physics of acoustic-vortical waves is
closely related to the two singularities of the gener-
alised Bessel differential equation, at the origin and
infinity, that are examined more closely next. The
generalised Bessel differential equation has two pa-
rameters, namely the order νand the degree µ, and
have several other applications that should be stud-
ied separately. It can generalise the Bessel, Neumann
and Hankel functions changing some properties of the
original functions.
The singularities of the original and generalised
Bessel differential equations are only at the origin and
infinity; the similarities arise because in both cases the
singularity is regular at the origin, and the differences
because the singularity at infinity is irregular with dif-
ferent degree. The solutions of the generalised Bessel
differential equation around the regular singularity at
the origin, [32], has: (i) indices that are exponents of
the leading power depending only on the order; (ii)
recurrence relation for the coefficients of the power
series expansion depending not only on the order, but
also on the degree. From (i) it follows the familiar
situation that generalised Bessel functions specify the
general integral for non-integer degree, and gener-
alised Neumann functions are needed for integer de-
gree. From (ii) it follows that the series expansion for
the generalised Bessel and Neumann functions differ
from the original series in having finite products mul-
tiplying each term; these finite products can be ex-
pressed as ratios of Gamma functions, whose argu-
ments become singular for zero degree. However, the
formulas of generalised Bessel and Neumann func-
tions in the form of ascending power series are not
well adapted for numerical computation when the in-
dependent variable is large because the series con-
verge slowly and an observation to their initial val-
ues offer no conclusion to the convergent values of
Jµ
ν(z)and Yµ
ν(z), [5], [32]. Therefore, the aim of
this work is to determine a formula which calculates,
in an easier way, the numerical values of the solution
of generalised Bessel equation when zis large and to
compare the results obtained with the original Hankel
functions.
Since the generalized Bessel differential equation
has an extra parameter that is dominant when the de-
gree is not zero, the asymptotic solutions are quite
different from those of the original Bessel differen-
tial equation because the irregular singularity at in-
finity is of degree two and not one as in the origi-
nal Bessel differential equation. The original Han-
kel functions scale asymptotically as an exponential
of the variable, and are oscillating for real variable
and monotonic for imaginary variable. The leading
term of the asymptotic solutions of the generalised
Bessel equation is monotonic (subsection 2.1) both
for real and imaginary variable: (i) there is one so-
lution without exponential factor, designated gener-
alised Hankel function of first kind; (ii) all other so-
lutions involve the generalised Hankel function of the
second kind (subsection 2.2) that scales an an expo-
nential of the square of the variable. To obtain a so-
lution of the generalised Bessel differential equation,
the Frobenius-Fuchs, [33], [34], method can be used
to specify power series around the regular singular-
ity at the origin and also can be used the normal in-
tegrals, [35], to specify asymptotic expansions in the
neighbourhood of the irregular singularity at the in-
finity (subsection 2.3).
When the degree tends to zero, the generalised
Hankel functions of first and second kinds do not con-
verge to the original functions for zero degree (sub-
section 3.1), in contrast with the generalised Bessel
and Neumann functions that do tend to their original
functions. The reason is that the origin is a regular sin-
gularity both for the original and generalised Bessel
differential equations, whereas the point at infinity is
an irregular singularity of different degree, namely 1
for the original and 2 for the generalised Bessel differ-
ential equation (subsection 3.2). It can be shown that
it is impossible to obtain an asymptotic solution of the
generalised Bessel differential equation that is contin-
uous and converges to the original Hankel function
after taking the limit of the degree to zero (subsection
3.3). The degree appears as a Hopf-type bifurcation
in the asymptotic solution of the generalised Bessel
differential equation because the solution has differ-
ent behaviour depending on the sign for real value of
degree: for negative degree, the solution is decaying,
for zero degree the solution is oscillatory and for pos-
itive degree the solution becomes divergent.
2 Asymptotic solutions of the
generalised Bessel differential
equation
The origin is a regular singularity of the generalised
Bessel differential equation and consequently the so-
lutions are ascending power series which are conver-
gent, with some terms having logarithms for integer
degree, [32]. Otherwise, the point at infinity of the
generalised Bessel differential equation is an irregu-
lar singularity, hence the solution (subsection 2.3) is
a linear combination of generalised Hankel functions
of two kinds; they consist on descending asymptotic
expansions and for generalised Hankel functions of
first kind they do not have an exponential factor (sub-
section 2.1) while the generalised Hankel functions
of second kind have an exponential factor (subsection
2.2).
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2.1 Asymptotic series for the Hankel
function of the first kind
The generalised Bessel differential equation is de-
fined in order to proceed to its asymptotic expansions
around the point at infinity.
Definition 1. Having the complex order ν∈Cand
degree µ∈Cas parameters and with the independent
variable z∈Calso complex, the generalised Bessel
differential equation is defined by
z2Q′′ +z1−µ
2z2Q′+z2−ν2Q= 0.(1)
Remark 1.The original Bessel differential equation
corresponds to zero degree with µ= 0.
The equation (1) has only two singularities: one at
the origin and the other at the infinity. The asymp-
totic solutions around the singularity at infinity are
obtained next.
Definition 2. Th generalised Bessel differential equa-
tion has an asymptotic solution as an expansion
around the point at infinity that corresponds to the in-
version of the origin:
ζ=1
z.(2a)
Defining Φ(ζ)≡Q(z), the differential equation (1)
leads to
ζ2Φ′′ +ζ1 + µ
2ζ2Φ′+1
ζ2−ν2Φ = 0.(2b)
Remark 2.If the point ζ= 0 of the equation (2b) is
a regular singularity then the point at infinity z=∞
of the generalised Bessel differential equation (1) is
also a regular singularity. However, since the factors
in curved brackets have double poles and are not ana-
lytic at ζ= 0, this last point is not a regular singularity
like the point z=∞. Thus, two linearly independent
solutions of (2b), assuming they are ascending power
series of ζor equivalently descending power series of
z, cannot exist in the form
Φϑ(ζ) =
∞
j=0
dj(ϑ)ζj+ϑ=
∞
j=0
dj(ϑ)z−j−ϑ=Q(z),
(3)
where the coefficients djdepend on ϑ. However, it is
possible that (i) one power series solution (3) exists at
most or (ii) maybe even none. It is demonstrated next
that the former case (i) is the correct option.
Theorem 1. The asymptotic solution of the gener-
alised Bessel differential equation (1) generalises the
Hankel function,
H(1)
µ, ν (z)∼z2/µ
1 +
N−1
j=1 −µz2/4−j
j!
×
j−1
l=0 l−1
µ2
−ν2
4+Oz−2N
,
(4)
which is of the first kind, valid for non-zero degree,
µ= 0, because the descending asymptotic expansion
has not an exponential factor.
Proof. The first series of (3) is substituted in (2b) or
the second series of (3) in (1). Both substitutions re-
sult to the same recurrence equation for the coeffi-
cients:
(j+ϑ)2−ν2dj(ϑ)
+µ
2(j+ϑ+2)+1dj+2(ϑ) = 0.(5)
If j=−2the indicial equation is ϑ=−2/µwhich
has only one root resulting in
d2j−2
µ
=−4(j−1−1/µ)2−(ν/2)2
µj d2j−2−2
µ.
(6a)
Applying the relation (6a) n-times leads to
d2j−2
µ=(−4/µ)j
j!
j−1
l=0 l−1
µ2
−ν2
4
(6b)
assuming
d0−2
µ= 1.(6c)
After substituting (6b) in (3) the first solution (4) in
the neighbourhood of the irregular singularity at in-
finity of the generalised Bessel differential equation
(1) is obtained.
Corollary 1. Replacing the products on the right-
hand side (r.h.s.) of (4) by
j−1
l=0 l−1
µ±ν
2=Γ(j±ν/2 −1/µ)
Γ(±ν/2 −1/µ),(7)
the generalised Hankel function can also be written
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as
H(1)
µ, ν (z)∼z2/µ
1 + d0(µ, ν)
N−1
j=1 −µz2/4−j
j!
×Γj+ν
2−1
µΓj−ν
2−1
µ
+Oz−2N
,(8a)
choosing a different leading constant factor:
d0(µ, ν) = Γν
2−1
µΓ−ν
2−1
µ−1
.(8b)
The solution (4), or equivalently (8a), is called the
generalised Hankel function of the first kind. This
function has coefficients increasing with jbecause
dj+2/dj∼O(j)in (6a), therefore the expansion is
not convergent as a series when j→ ∞; nonetheless,
it is an asymptotic expansion that can be evaluated
with a finite number of terms, j < ∞, as |z|→∞.
To guarantee the convergence of the Frobenius se-
ries, [36], the Fuchs theorem, [37], cannot be applied
around an irregular singularity because the theorem
in this case leads to an asymptotic expansion and not
a convergent series. When |z| → ∞, the gener-
alised Hankel function of the first kind (4) decays for
ℜ(µ)<0and diverges for ℜ(µ)>0. The terms
−µz2/4 exist in these Hankel functions (4); for real
zthe term is negative and so the factor −µz2/4−j
has alternating sign, otherwise for pure imaginary z
the term is positive and consequently the same factor
is positive for any j.
The Fig. 1 shows the effect of increasing the num-
ber of terms in the series expansion of the gener-
alised Hankel function of the first kind and, for val-
ues greater than N= 10, the new terms of summa-
tion don’t lead to noticeable difference because the
term (µz2/4)−j/j!decreases for greater values of j.
Therefore, N= 15 is established for all other plots.
The Fig. 1 also shows, for N= 15, the differences
induced by varying the value of ν. For all values of ν
and µ, the Hankel function “explodes” at z= 0 (this
effect is visible in the plot for ν= 15) and for that rea-
son the plots don’t show the function for values near
the origin (also, the objective is to study the behav-
ior of functions for large values through asymptotic
expansions). The greater the value of the parameter
ν, the greater the value of generalised Hankel func-
tion is, for all values of z. In the Fig. 1, all the plots
diverge because µis a real positive number and the
greater the value of νthe more quickly the function
diverges. The Hankel function is even with respect to
10 12 14 16 18 20
z
100
150
200
250
300
350
400
450
H(1)
µ,ν(z)
N= 0
N= 5
N= 10
N= 15
10 12 14 16 18 20
z
0
100
200
300
400
500
600
700
800
900
H(1)
µ,ν(z)
ν= 0
ν= 5
ν= 10
ν= 15
Fig. 1: Generalised Hankel function of the first kind,
setting µ= 1 and ν= 5, for different values of the
number of terms in the series expansion (left), or set-
ting µ= 1 and N= 15, for different values of ν
(right).
zand that is the reason why there is no need to illus-
trate the function for negative values of z(the plots
are symmetric regarding the vertical axis) and is also
even with respect to ν, that is, the plots are exactly
the same for symmetric values of ν(for instance, the
plots obtained with ν= 5 and ν=−5are identical).
All the parameters and values of the function are real
values in the Fig. 1 to not appear any complex num-
bers and to illustrate the function in a single 2D-plot
to make easier the observations, and because the pa-
rameters are usually real in the differential equations
deduced from the physics subjects.
The Fig. 2 shows the strong dependence of the val-
ues of the generalised Hankel function of the first kind
on the parameter µ. The generalised Hankel function
is not valid for µ= 0. It diverges for real positive val-
ues of µ, but decays for negative real values and the
step of increasing or decreasing is higher for lower
values of µ. As previously in the Fig. 1, to facilitate
the comparison of plots resulting from different con-
ditions, all the parameters and the variables are real
numbers to not plot complex numbers. Again, the
plots show that the function always diverges at z= 0
for all values of µ. Lastly, the Hankel function is even
with respect to zso the Fig. 2 only illustrates the func-
tion for positive values of z.
2.2 Asymptotic expansion for the
generalised Hankel function of the
second kind
The factor exp [A(z)], where A(z)is a polynomial of
z, has an essential singularity at infinity, [38], [39],
the asymptotic scaling of the original Hankel func-
tions, [40], is
H(1,2)
ν(z)∼2
πz exp ±iz−νπ
2−π
4.(9)
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0 200 400 600 800 1000
z
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
H(1)
µ,ν(z)
µ=−5
µ=−7
µ=−9
µ=−11
0 200 400 600 800 1000
z
0
2
4
6
8
10
12
14
16
H(1)
µ,ν(z)
µ= 5
µ= 7
µ= 9
µ= 11
Fig. 2: Generalised Hankel function of the first kind,
setting N= 15 and ν= 1, for different negative
values of the parameter µ(left) or positive values of
the parameter µ(right).
When ζ→0the coefficient of Φ′in (2b) is analytic if
µ= 0, but has a double pole if µ= 0. Consequently,
in spite of the original Hankel function correspond-
ing to the generalised Hankel function with zero de-
gree, when the degree tends to zero µ→0the limit
is not continuous. The degree of the irregular singu-
larity at infinity in the generalised Bessel differential
equation (1) is two, as will be proved next, and there-
fore higher than the degree of the singularity at infin-
ity in the original Bessel differential equation which
is one. Consequently the limit when µ→0is descon-
tinouous (section 3).
The generalised Bessel differential equation (1)
has an asymptotic solution which is linearly indepen-
dent of the generalised Hankel function of first kind
(4). The reason is that the asymptotic solution can-
not be written as in (3) because: (i) the corresponding
indicial equation, ϑ=−2/µ, has only one solution;
(ii) if there is such a solution then the point at infinity
would be a regular singularity instead of an irregu-
lar singularity. Nonetheless, an essential singularity
must be present in the asymptotic solution of the gen-
eralised Bessel differential equation (1) linearly inde-
pendent from the generalised Hankel function of first
kind. The essential singularity can be written in the
form of a normal integral, [41].
Φ(ζ) = exp A1
ζΨ(ζ).(10a)
Ihe degree of the polynomial A(z)determines the de-
gree of the essential singularity at infinity, provided
that the remaining factor Ψis a Frobenius series or at
least an asymptotic expansion:
Ψχ(ζ) =
∞
j=0
ej(χ)ζj+χ.(10b)
The coefficients ejdepend on χ. This leads to the
theorem 2.
Theorem 2. An asymptotic solution of the gener-
alised Bessel differential equation (1) is the gener-
alised Hankel function of the second kind
H(2)
µ,ν (z)∼exp µz2
4z−2−2/µ
×
1 +
N−1
j=1 µz2/4−j
j!
×
j
l=1 l+1
µ2
−ν2
4+Oz−2N
(11)
consisting of a descending asymptotic series multi-
plied by an exponential term. This series solution ex-
ists only for non-zero degree, µ= 0.
Proof. The asymptotic normal integral (10a) is sub-
stituted in (2b) leading to
ζ2Ψ′′ +ζ1 + µ
2ζ2+ 2A′ζΨ′
+ζ2A′2+ζ2A′′ +ζA′1 + µ
2ζ2
+1
ζ2−ν2Ψ = 0.(12)
to obtain a second solution of the generalised Bessel
differential equation (1) that, unlike the first (4), is a
normal integral, and hence linearly independent. The
polynomial A′in (12) should be chosen such that the
solution for Ψin the form (10b) exists. The simplest
choice is an inverse power A′∼ζ−n. For n= 1, the
relation A′∼ζ−1leads to A∼ϑln ζwhich is equal
to eA∼ζϑbeing this the Frobenius solution (3). For
n= 2, the relation A′∼ζ−2leads to A∼ζ−1and
eA∼exp(1/ζ)∼exp z. This case corresponds to
zero degree µ= 0 and leads to the asymptotic scal-
ing of the original Hankel function (9) and therefore
a solution of the original Bessel differential equation
that has an irregular singularity at infinity of degree
one. It follows that the generalised Bessel differential
equation with non-zero degree has a stronger irregu-
lar singularity at infinity and hence the lowest degree
must be two leading to A∼ζ−2or A′∼ζ−3. To
confirm the degree of the irregular singularity at in-
finity of the generalised Bessel differential equation,
the equality
A′1
ζ=b
ζ3(13a)
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is substituted in (12) leading to the differential equa-
tion (13b)
ζ2Ψ′′ +ζ1 + 2b+µ/2
ζ2Ψ′
+b(b+µ/2)
ζ4+1−2b
ζ2−ν2Ψ = 0.
(13b)
The fourth-order pole in the square bracketed term
can be eliminated by choosing
b=−µ
2(14a)
and consequently the differential equation (13b) is
simplified to
ζ2Ψ′′ +ζ1−µ
2ζ2Ψ′+1 + µ
ζ2−ν2Ψ = 0.
(14b)
It is impossible to have two linearly independent so-
lutions of the form (10b) because the point at infinity
is still an irregular singularity of (14b). Nonetheless,
knowing already one asymptotic solution (4) of the
generalized Bessel differential equation (1), only one
more solution is needed.
Substituting of (10b) in (14b) results in the recur-
rence formula
(j+χ)µ
2−1ej+2(χ) = (j+χ)2−ν2ej(χ).
(15)
When j=−2, the indicial equation is
µχ
2−1−1e0(χ) = 0,(16a)
and noting that e0(χ)= 0, it has one root:
χ= 2 + 2
µ.(16b)
Assuming that
e02 + 2
µ= 1 (17a)
and substituting (16a) in (15), the coefficients of the
asymptotic expansion, needed for the equation (10b),
are specified:
e2j2 + 2
µ=(2j+ 2/µ)2−ν2
µj e2j−22 + 2
µ
=(µ/4)−j
j!
j
l=1 j+1
µ2
−ν2
4.
(17b)
The equation above is substituted first in the Frobe-
nius series (10b) and after in the asymptotic solution
(10a) that is multiplied by an exponential term. Re-
garding (13a) and (14a), the argument of the exponen-
tial satisfies
A′=−µ
2ζ3,(18a)
implying
A=µ
4ζ2(18b)
and consequently
exp A1
ζ=exp µ
4ζ2=exp µz2
4.
(18c)
Substituting (17b) and (18c) in (10b) and (10a) re-
spectively leads to the second asymptotic solution
of the generalised Bessel differential equation that is
called the generalised Hankel function of the second
kind (11).
Corollary 2. If in the last term on the r.h.s. of (11) a
pair of relations similar to (7) is used, but with −1/µ
replaced by 1/µ, an alternate expression for the gen-
eralised Hankel function of second kind is obtained,
specifically
H(2)
µ,ν ∼exp µz2
4z−2−2/µ
1 + e0(µ, ν)
×
N−1
j=1 µz2/4−j
j!Γj+1+ν
2+1
µ
×Γj+ 1 −ν
2+1
µ+Oz−2N
.
(19a)
The leading constant factor e0is replaced by
e0(µ, ν) = Γ1 + ν
2+1
µΓ1−ν
2+1
µ−1
.
(19b)
The Fig. 3 illustrates the generalised Hankel func-
tion of the second kind, setting again N= 15 and
µ= 1, and leading to the same conclusions as in the
Fig. 1. However, in opposition to the example of Fig.
1, the values of the function increase more quickly for
lower values of ν. But the main difference is that, for
the same value of z, the function of the second kind
reaches greater values than the function of the first
kind, due to the exponential function that is present
only in the function of the second kind.
The Fig. 4 illustrates the generalised Hankel func-
tion of the second kind, setting N= 15 and ν= 1,
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10 12 14 16 18 20
z
0
2
4
6
8
10
12
14
16
18
H(2)
µ,ν(z)
×1037
ν= 0
ν= 5
ν= 10
ν= 15
10 10.2 10.4 10.6 10.8 11
z
0
2
4
6
8
10
12
H(2)
µ,ν(z)
×108
ν= 0
ν= 5
ν= 10
ν= 15
Fig. 3: Generalised Hankel function of the second
kind, setting µ= 1 and N= 15, for different val-
ues of ν, plotted in the range 10 ≤z≤20 (left) or in
the range 10 ≤z≤11 (right).
Fig. 4: Generalised Hankel function of the second
kind, setting N= 15 and ν= 1, for different nega-
tive values of the parameter µ(left) or positive values
of the parameter µ(right).
plotted for different values of µ. The observations are
the same as the Fig. 2, however there is two big dif-
ferences: for the same values of the parameters and z,
the function of the second kind reaches greater values
than the function of the first kind, due to the presence
of an exponential function in the Fig. 4; the parameter
µappears in the exponential term of the Hankel func-
tion of the second kind and consequently the values
of that function are strongly dependent of the param-
eter µ, more than in case of the function of the first
kind. As it can be seen from the Fig. 4, for z= 3.5,
changing the value of µonly 0.5 units (for instance,
from 3.5 to 4) can lead to the difference in the Hankel
function of 40000 units.
2.3 Wronskian for linearly independent
asymptotic solutions and general
integral
It is shown next that the Wronskian between the gen-
eralised Hankel functions of first (4) and second (11)
kinds is non-zero if the degree is also non-zero. In
that case, both asymptotic solutions of the generalised
Bessel differential equation (1) are linearly indepen-
dent and their linear combination determines the gen-
eral integral. A preliminary lemma on the Wronskian
of any two solutions of the Bessel differential equa-
tion, [32], is used in a subsequent lemma to specify
the Wronskian of the generalised Hankel functions of
two kinds.
Lemma 3. Two particular solutions (Q1)and (Q2)of
the generalised Bessel differential equation (1) have
the Wronskian
W[Q1(z), Q2(z)] ≡Q1(z)Q′
2(z)−Q′
1(z)Q2(z)
=W(0)
zexp 1
4µz2.(20)
Lemma 4. The Wrosnkian of the generalised Han-
kel functions of first (4) and second (11) kinds, for
non-zero degree, µ= 0, has the asymptotic expan-
sion equal to
WH(1)
µ,ν (z), H(2)
µ,ν (z)∼µ
2zexp 1
4µz2.(21)
The two functions are linearly independent because
the Wronskian is non-zero.
Proof. The asymptotic solutions (4) and (11) have the
leading terms
H(1)
µ,ν (z)∼z2/µ(22a)
and
H(2)
µ,ν (z)∼z−2−2/µexp 1
4µz2,(22b)
leading to the Wronskian (21) that is equal to (20) if
W(0) = µ
2.(22c)
From this follows immediately the next theorem.
Theorem 5. The general asymptotic solution (2b) of
the generalised Bessel differential equation (1) is a
linear combination with arbitrary constants (C1)and
(C2)of the generalised Hankel functions of first (4)
and second (11) kinds,
Q(z)∼C1H(1)
µ,ν (z) + C2H(2)
µ,ν (z).(23)
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This linear combination is valid for any value of de-
gree, including zero degree corresponding to original
Hankel functions.
The general solution of the generalised Bessel dif-
ferential equation (1) in the neighbourhood of the reg-
ular singularity at the origin for non-integer order in-
volves generalised Bessel functions while for integer
order the Neumann functions are present in the solu-
tion, [32]. In both cases, the solutions hold for finite
z. Hence, both solutions overlap with the generalised
Hankel function that also holds for non-zero degree
µand large variable z. Consequently, it follows the
theorem 6.
Theorem 6. The asymptotic expansion of the gener-
alised Bessel functions with non-zero degree, µ= 0,
has the form
Jµ
ν(z) = D+(µ, ν)H(1)
µ,ν (z) + D−(µ, ν)H(2)
µ,ν (z).
(24)
Furthermore the coefficients D+(µ, ν)and D−(µ, ν)
are functions of the degree µand order νsatisfying
the relations
−4sin(πν)
πµ =D+(µ, ν)D−(µ, −ν)
−D+(µ, −ν)D−(µ, ν).(25)
Proof. The general integral (23) for the generalised
Bessel function applies to any order and non-zero de-
gree. Hence the generalised Bessel function must
be expressible in the form (24) with coefficients
D+(µ, ν)and D−(µ, ν)determined by comparing
the solutions around the origin for |z|<∞and
around the point at infinity for |z|>0in the region
of overlap 0<|z|<∞. In the situation where
mu = 0 corresponding to the original Bessel func-
tions, the asymptotic expansion is typically found us-
ing an integral representation or using the method of
Wronskians, [42], from the asymptotic solution of the
original Bessel differential equation. The asymptotic
solutions (4) and (11) of the generalised Hankel func-
tions are the same when changing the sign of the order
nu,
H(1,2)
µ,−ν(z) = H(1,2)
µ,ν (z),(26a)
and therefore
Jµ
−ν(z) = D+(µ, −ν)H(1)
µ,ν (z)+D−(µ, −ν)H(2)
µ,ν (z).
(26b)
Using (24) and (26a) the Wronskian of the generalised
Bessel functions of orders ±νis related to the Wron-
skian of the Hankel functions of two kinds and order
ν, with all four functions having the same degree µ:
WJµ
+ν(z), Jµ
−ν(z)
= [D+(µ, ν)D−(µ, −ν)−D+(µ, −ν)D−(µ, ν)]
×WH(1)
µ,ν (z), H(2)
µ,ν (z);(27)
using the Wronskian of the generalised Bessel func-
tions (lemma 2 in [32])
WJµ
+ν(z), Jµ
−ν(z)=−2
πz sin(νπ)exp 1
4µz2
(28)
and of the generalised Hankel functions of first and
second kinds (21) in (27) demonstrates (25).
Thus for non-integer order the coefficients relating
(24) the initial and asymptotic solutions of the gener-
alised Bessel equation must satisfy the relation (25).
Therefore, the method of Wronskian leads to only one
relation between the terms D+(µ, ν)and D−(µ, ν).
The determination of both coefficients uses instead
the relation between generalised Bessel and conflu-
ent hypergeometric functions presented in a follow-
on paper.
The general solution of the generalised Bessel dif-
ferential equation (1) for integer order includes gen-
eralised Bessel functions
Jµ
n(z) = D+(µ, n)H(1)
µ,n(z) + D−(µ, n)H(2)
µ,n(z)
(29a)
and generalised Neumann functions
Yµ
n(z) = E+(µ, n)H(1)
µ,n(z) + E−(µ, n)H(2)
µ,n(z),
(29b)
where the coefficients E+(µ, n)and E−(µ, n)are
functions of the complex degree µand integer order
n, and whose asymptotic forms are similar to (24) be-
cause the latter holds for all orders, leading therefore
to the next theorem.
Theorem 7. For the same integer order ν=nand
complex non-zero degree µ= 0, the generalised Han-
kel function of the first (4) and second (11) kinds are
linear combinations of generalised Bessel (29a) and
Neumann (29b) functions with coefficients related by:
D+(µ, n)E−(µ, n)−D−(µ, n)E+(µ, n)
=4C
πµ = 4(−1)n
πµ−1−nΓ(−n/2 −1/µ)
Γ(n/2 −1/µ).(30)
Proof. From (29a) and (29b) follows the relation be-
tween the Wronskians:
W[Jµ
n(z), Y µ
n(z)]
= [D+(µ, n)E−(µ, n)−E+(µ, n)D−(µ, n)]
×WH(1)
µ,n(z), H(2)
µ,n(z).(31)
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The generalised Bessel and Neumann functions of ar-
bitrary degree and integer order (lemma 3 in [32])
have the Wrosnkian
W[Jµ
n(z), Y µ
n(z)] = 2
πz C(µ)exp 1
4µz2(32a)
involving the coefficient
1
C(µ)≡
n−1
l=0 1−µl−n
2
= (−µ)n
n−1
l=0 l−n
2−1
µ
= (−µ)nΓ(n/2 −1/µ)
Γ(−n/2 −1/µ).(32b)
Replacing (32b) and (21), valid for µ= 0, in (31)
proves (30).
The method of Wronskian leads to one relation
between the four coefficients - D+(µ, n),D−(µ, n),
E+(µ, n)and E−(µ, n)- whereas the use of confluent
hypergeometric functions leads to explicit functions
of E+(µ, n)and E−(µ, n)as shown in a subsequent
paper.
3 Bifurcation of the asymptotic
solutions around the point at
infinity for zero degree
The reason the asymptotic solutions of the generalised
Bessel equation and the original Bessel equation stop
working as the degree approaches to zero is because
the essential singularity at infinity changes from de-
gree two to degree one. It will be shown in two other
ways that a solution that is continuous in µnear the
origin cannot exist (subsection 3.2). This is like a
Hopf-type bifurcation because the asymptotic solu-
tion changes its behaviour depending on the sign of
degree: if the degree is negative, the solution decays,
if the real degree is zero the solution is oscillatory and
finally if the degree is positive the solution diverges.
(See section 3. 3)
3.1 Discontinuity in the degree between the
original and generalised Hankel
functions
The singularity at the origin of the generalised Bessel
differential equation (1) is regular regardless the value
of degree µ. The indices are determined only by the
order νwhile the degree µappears only in the coef-
ficients of the Frobenius-Fuchs series expansion. As
a consequence, when the degree approaches the value
zero, the generalised Bessel (33a) and Neumann (33b)
functions tend respectively to the original Bessel and
Neumann functions:
lim
µ→0Jµ
ν(z) = Jν(z); (33a)
lim
µ→0Yµ
n(z) = Yn(z).(33b)
The situation is different when considering the
asymptotic solutions near the irregular singularity at
infinity: (i) the generalised Hankel function of first
(4) and second (11) kinds become invalid when the
degree is zero; (ii) the original Hankel functions of
two kinds (9) have an essential singularity of degree
one, whereas the degree of the singularity in the gen-
eralised Hankel function of second kind (11) is two
while the generalised Hankel function of first kind (4)
does not have an essential singularity. So it is not pos-
sible to satisfy the continuity in the parameter µ:
lim
µ→0H(1,2)
µ,ν (z)=H(1,2)
ν(z).(34)
This could be predicted from the generalised Bessel
differential equation (1) since the degree µappears
only in the factor µz3Q′: (i) when z→0, this term
tends to zero and therefore preserves the continuity
of the ascending power series specifying the gener-
alised and original Bessel functions (33a), and like-
wise for the generalised and original Neumann func-
tions (33b), that involve a similar logarithmic term;
(ii) as z→ ∞, the term µz3Q′= 0 vanishes for
µ= 0, but diverges for µ= 0 because µz3Q′→ ∞,
and thus the original and generalised Hankel func-
tions are discontinuous, as stated in (34), and are
represented by asymptotic expansions in descending
powers of z.
The solutions of the generalised and original
Bessel differential equations are: (i) continuous with
regard to the degree near the origin as z→0because
µdoes not appear to leading order in the ascending
power series solutions near the regular singularity at
the origin; (ii) discontinuous with regard to the de-
gree asymptotically at infinity as z→ ∞ because µ
appears in the leading term of the descending power
asymptotic expansions and normal integrals near the
irregular singularity at infinity that has degree 1 for
µ= 0 and degree 2 for µ= 0. Concerning the
ascending power series solutions around the regular
singularity at the origin only the Neumann but not the
Bessel function has a logarithmic term. Concerning
the descending power asymptotic expansions near the
irregular singularity at the point at infinity, there is an
essential singularity associated with the exponential
factor in the normal integral in all cases, including the
generalised Hankel functions of the second kind, with
the exception being only the generalised Hankel func-
tion of the first kind, in which no such factor appears.
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This is not the case for the original Hankel functions
of two kinds, which are asymptotic solutions of the
original Bessel differential equation, specified by a
normal integral with essential singularity of degree
unity, leading to oscillatory solutions for real variable.
In contrast, the generalised Bessel equation with non-
zero degree has an asymptotic solution (23) with an
essential singularity of degree two (18c), leading to a
monotonic response for real µand z. The irregular
singularities of the original (equation (9)) and gen-
eralised (equations (4) and (11)) Bessel differential
equations are of different degree, specifically one and
two for the original and generalised Bessel equations,
and consequently the two differential equations have
distinct asymptotic expansions. This shows that the
solution of an ordinary differential equation around a
singularity may be discontinuous with regard to a pa-
rameter, in this example the degree of the generalised
Bessel differential equation.
The main difference between the original, µ= 0,
and generalised, µ= 0, Bessel equations is the ir-
regular singularity at infinity. In the original equa-
tion, the singularity has degree one, while in the gen-
eralised equation, it has degree two. TThis change
is shown in the way the original Hankel functions
(9) scale asymptotically compared to the generalised
Hankel functions of second kind (11). The question
could be raised if there is one or two asymptotic solu-
tions of the generalised Bessel equation (9) such that:
(i) are valid for all values of the degree µ; (ii) for non-
zero values of the degree, µ= 0, the asymptotic so-
lution scales as in (18c); (iii) for zero degree, µ= 0,
when the asymptotic factor (18c) is equal to one, the
asymptotic factor is given by e±izas for the original
Hankel functions (9). It will be proved in more than
one way that it is impossible to meet all the condi-
tions (i) to (iii). Consequently, there is no solution in
the previous form.
The original and generalised Hankel functions are
distinct and they demonstrate a discontinuity with re-
gard to the degree µof the same differential equation;
the irregular singularity at infinity is of different de-
gree – two for non-zero degree and one for zero de-
gree in the generalised Bessel differential equation (1)
– leading to the discontinuity of the asymptotic so-
lution. The common factor (18c) that appears in all
Wronskians – specifically in the equations (21), (28)
and (32a) – between all pairs of linearly independent
solutions leads to asymptotic expansions distinct from
those of the original Bessel equation (9). It will be
proved next that it is impossible to have one or two
asymptotic solutions of the generalised Bessel differ-
ential equation that become the original Hankel func-
tions when the degree tends to zero. Two independent
ways (subsections 3.2 and 3.3) will be used to prove
the preceding statement.
3.2 Non-existence of an asymptotic normal
integral continuous for zero degree
Theorem 8. The generalised Bessel differential
equation (1) with non-zero degree does not have an
asymptotic solution in the form of a normal integral
(10a) continuous with respect to the degree µas it ap-
proaches to zero.
Proof. Starting with the generalised Bessel differen-
tial equation (1), to seek a solution as a first normal in-
tegral (10a) with leading term (18c) leads to the differ-
ential equation (14b). When the degree is zero, µ= 0,
the leading term (18c) is equal to one, and it possible
that the normal integral for the differential equation
(14b) would result in the original Hankel functions
(9). To find out if this is possible, a second normal
integral is assumed to be part of the solution of the
differential equation (14b),
Ψ(ζ) = eB(1/ζ)Θ(ζ),(35a)
so that multiplied with the first normal integral, stated
in (13a), the asymptotic solution of the generalised
Bessel differential equation is equal to
Φ(ζ) = eA(1/ζ)eB(1/ζ)Θ(ζ)(35b)
where the function Θmust satisfy the following dif-
ferential equation
ζ2Θ′′ +ζ1−µ
2ζ2+ 2B′ζΘ′
+ζ2B′2+B′′+ζ−µ
2ζB′
+1 + µ
ζ2−ν2Θ = 0,(36)
obtained substituting (35a) in (14b).
The first normal integral was found with a triple
pole for A′, as in (13a); an asymptotic expansion with-
out essential singularity can be obtained with a simple
pole for B′. For that reason B′should have a double
pole,
B′=g
ζ2,(37a)
leading by substitution in (36) to the differential equa-
tion
ζ2Θ′′ +ζ1−µ
2ζ2+2g
ζΘ′
+−µg
2ζ3+1 + µ+g2
ζ2−g
ζ−ν2Θ = 0.
(37b)
The term in square brackets has a triple pole, that
could be suppressed by setting g= 0 resulting in a
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trivial differential equation B′= 0 in (37a). This
may demonstrate that trying to find the normal inte-
gral might be impossible.
The arbitrary constant gin (37a) can be specified
to suppress, in the differential equation (37b), the dou-
ble pole in square brackets in the coefficient of Θ, by
setting
g2+1+µ= 0; (38a)
the roots
g=∓i1 + µ(38b)
lead, using the equation (37a), to
B=−g
ζ=±i√1 + µ
ζ.(38c)
Using (18b) and (38c) in (37b) leads to
Φ(ζ) = exp µ
4ζ2±i√1 + µ
ζΘ(ζ)
=exp 1
4µz2exp ±iz1 + µΘ1
z
(39)
where: (i) the dominant term is (18c) as for the gener-
alised Hankel function of second kind (11) with non-
zero degree, µ= 0; (ii) the next leading term reduces
to exp(±iz)for zero degree, µ= 0 (in this case, the
leading term is unity), and actually coincides with the
asymptotic scaling of the original Hankel functions
(9); (iii) the function Θ(ζ)must be an asymptotic ex-
pansion in Frobenius form,
Θϕ(ζ)∼
∞
j=0
fj(ϕ)ζϕ+j=
∞
j=0
fj(ϕ)z−ϕ−j,(40)
so that the function Φin (39) represents the two
asymptotic solutions of the generalised Bessel differ-
ential equation. To make this possible, only one index
ϕmust exist; since ζ= 0 is an irregular singularity
of the differential equation (37b), there cannot be two
indices.
Substituting (38b), the differential equation (37b)
becomes
ζ2Θ′′ +ζ1−µ
2ζ2∓2i√1 + µ
ζΘ′
−ν2∓i
ζ1 + µ
2ζ21 + µΘ = 0.
(41)
To obtain the coefficients, the substitution of (40) in
the equation (41) leads to the following recurrence
equation:
±iµ
21 + µfj+3(ϕ)−µ
2(ϕ+j+ 2)fj+2(ϕ)
∓i1 + µ(2j+ 2ϕ+ 1)fj+1(ϕ)
+(ϕ+j)2−ν2fj(ϕ) = 0.(42)
Setting j=−3leads to
µ1 + µf0(ϕ) = 0,(43a)
and from f0(ϕ) = 0 follows fj(ϕ) = 0, showing that
there is no index ϕand only a trivial solution exists:
Θ(ζ) = 0.(43b)
The reason for the existence of only a trivial solution
(43b) is that the indicial equation has no roots, be-
cause it is specified by the triple pole in the coeffi-
cient of Θin the differential equation (37b). Conse-
quently, it has been shown that it is impossible to find
an asymptotic solution of the generalised Bessel dif-
ferential equation (1) when the degree µis not zero
in such a way that provides also an asymptotic solu-
tion of the original Bessel differential equation after
setting the limit µ→0.
3.3 Alternative proof of discontinuity of
Hankel functions for zero degree
It is possible to use the method of “reductio ad absur-
dum” to demonstrate in an independent and simple
way that the generalised Bessel differential equation
has no solution of the form (39).
Theorem 9. There is no asymptotic solution of the
form (35a) and (35b) involving a descending asymp-
totic expansion (40) which satisfies the generalised
Bessel differential equation (1).
Proof. The proof is made by “reductio ad absurdum”.
Let’s assume that there is a solution like (39) with Θ
being a descending asymptotic expansion as in (40).
The solution (39) has to be a linear combination of
generalised Hankel functions of first (4) and second
(11) kinds:
exp 1
4µz2±iz1 + µΘ1
z
=G+H(1)
µ,ν (z) + G−H(2)
µ,ν (z).(44)
The equation (44) shows that the r.h.s. only has even
powers of zwhile the left-hand side (l.h.s.) has both
even and odd powers of z. The even powers in the
r.h.s. come from the equations (4) and (11); the even
and odd powers in the other side come from the expo-
nential term. For that reason, the equality is impossi-
ble.
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Remark 3.If the exponential argument on the l.h.s.
did not have the second term, then both sides of the
equation would have only even powers of zand the
equality in (44) would be possible. However, that
case does not include the original Hankel functions.
To conclude: (i) the simplest asymptotic solu-
tion of the generalised Bessel differential equation (1)
is the generalised Hankel function of first kind (4),
which is the only solution that has no essential singu-
larity; (ii) any other solution has to include the expo-
nential factor (18c) leading to an essential singular-
ity, as in the case of the generalised Hankel function
of second kind (11); (iii) the general solution is there-
fore a linear combination of the preceding two and
thus cannot be of the form
Φ1
z=exp 1
4µz2exp [±izh(µ)] Θ 1
z,
(45)
that would be continuous with the original Hankel
functions; (iv) the discontinuity between the gener-
alised and original Hankel functions, stated in (34),
is associated with different asymptotic solution of the
original and generalised Bessel differential equation
that has at infinity an irregular singularity of different
degree, respectively 1 for µ= 0 in the former case
and 2 for µ= 0 in the latter case.
A very simple example of bifurcation of the solu-
tion of an ordinary differential equation is given by
Φ′′ +µΦ = 0,(46)
with real constant coefficient µ. The solution is: (i)
oscillatory,
Φ (x) = C1cos (νx) + C2sin (νx),(47a)
for positive µ > 0hence real µ; (ii) decaying or di-
verging,
Φ (x) = C1exp (νx) + C2exp (−νx),(47b)
for negative µ < 0; (iii) linearly diverging,
Φ (x) = C1+C2x, (47c)
in the intermediate case µ= 0 between (i) and (ii),
where ν≡ |µ|1/2 and (C1, C2)are arbitrary constants
of integration. The bifurcation at µ= 0 separates
sinusoidal oscillations in (47a) for µ > 0for all real
x, from decay or instability for µ≤0in (47b) and
(47c). For µ= 0 in (47c) there is linear instability.
For µ < 0in (47b): (i) for positive x > 0there is
exponential decay for C1= 0 =C2and exponential
instability for C1= 0; (ii) for negative x < 0there is
exponential decay for C1= 0 = C2and exponential
instability for C2= 0; (iii) for the full range of real
values of xthere is always exponential instability for
(C1, C2)= (0,0).
4 Conclusion
Since the generalised Bessel differential equation (1)
has only two singularities, one at the origin z= 0 and
the other at infinity z=∞, the method of expansion
in powers of zadopted in the present paper has the
advantage that the solutions apply over the full do-
main of the independent variable 0<|z|<∞out-
side the singularities z={0,∞}. Because the sin-
gularity is regular at the origin, the Frobenius-Fuchs
method, [36], [37], [39], [40], specifies power series
solutions convergent for |z|>0involving: (i) only
generalised Bessel functions of orders ±νand degree
µif νis not an integer; (ii) generalised Bessel and
Neumann functions of order νand degree µif νis an
integer. In both cases the original Bessel and Neu-
mann functions correspond to the case of zero degree
µ= 0. Another situation is for the solution in the
neighbourhood of the singularity at infinity because:
(i) it is an irregular singularity and the method of nor-
mal integrals leads to two Hankel functions specified
by asymptotic expansions that provide a good approx-
imation at order Nfor large |z|but do not converge
as N→ ∞; (ii) the irregular singularity at infinity is
of distinct degree – one for the original and two for
the generalised Bessel equations – leading to asymp-
totic expansions in powers of zwith negative inte-
ger exponents, that are multiplied by an exponential
of the form exp (αz)with constant αor exp βz2
with constant βrespectively for the original and gen-
eralised Hankel functions.
The asymptotic solution of the generalised Bessel
differential equation (1) is discontinuous when the de-
gree becomes zero. That discontinuity is like a Hopf-
type bifurcation because for the real variable zand
order ν, the asymptotic solution is: (i) oscillatory for
zero degree, µ= 0, the same behaviour as in the orig-
inal Hankel functions for zero degree (9); (ii) unstable
for positive degree, µ > 0since both the generalised
Hankel functions of first (4) and second (11) kinds
diverge as z→ ∞; (iii) stable for negative degree,
µ < 0, since both the generalised Hankel functions
of first (4) and second (11) kinds decay as z→ ∞.
The original Hopf bifurcation, [43], is about an au-
tonomous differential system relating to a non-linear
second-order differential equation with constant co-
efficients, [44]. The present example studies a linear
second-order differential equation with variable coef-
ficients (1) that can be transformed to an autonomous
system of two first-order differential equations. The
present method has proved that it is possible to get
explicit solutions of a differential equation around a
bifurcation with the method of normal integrals, [35],
[36], that generalises the method of regular integrals
with branch-points, [33], [34], [37].
The generalised Bessel differential equation can
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have the form of a confluent hypergeometric differen-
tial equation, leading to relations between the Kum-
mer functions, [45], [46], and the generalised Bessel,
Neumann and Hankel functions that will be explored
in a future work. The direct solutions of the gener-
alised Bessel differential equation given here tries to
keep as close as possible to the original Bessel, Neu-
mann and Hankel functions, to highlight more clearly
the differences associated with the generalization.
The generalised Bessel functions are an alternative
to the Kummer confluent hypergeometric functions,
[45], and the Whittaker functions, [39], with the ad-
vantage that the connection with the original Bessel
functions is much simpler: (i) the original Bessel
functions of order νare the generalised Bessel func-
tions of order νand degree µzero,
Jν(z) = J0
ν(z),(48)
and thus are identified by a single parameter with µ=
0; (ii) the relation between confluent hypergeometric
functions and the original Bessel functions is a limit,
Jν(z) = lim
a→∞ z
2νFa, ν;−z
a,(49)
that is much less obvious than (48); (iii) the relation
of Whittaker functions is
Jν(z) = 2−νzν/2 lim
a→∞ (−a)−ν/2 e−z/(2a)
×Wν/2−a,ν/2−1/2 −z
a,(50)
which is again a limit more complex than (48). Thus
the generalised Bessel functions are equivalent to
Kummer’s confluent hypergeometric functions and to
Whittaker functions that have the simplest relation
(48) to the original Bessel functions. The simple re-
lation (48) also applies to the generalised and original
Neumann functions,
Yν(z) = Y0
ν(z),(51)
since, like the generalised Bessel functions, they are
solutions of the generalised Bessel differential equa-
tion (1) around the origin where as a regular singular-
ity. The relations (48) and (51) do not extend to Han-
kel functions because the original/generalised Hankel
functions are asymptotic solutions of respectively the
original/generalised Bessel differential equation that
contains at infinity irregular singularities with differ-
ent degrees. The original Bessel functions have nu-
merous applications to problems in physics and engi-
neering problems, [4], [5], [40], [47], [48], involving
special functions, [39], [49], [50], [51], that are both
a classical subject, [52], [53], [54], [55], and a topic
of current research, [56], [57], [58], [59], some re-
cent examples concern, [60], [61], Bessel, [42], [62],
[63], [64], and Gaussian and confluent hypergeomet-
ric, [65], functions and various physical and engineer-
ing problems, [66], [67], concerning specifically the
generalised Bessel differential equation; solutions (i)
were obtained previously as Frobenius-Fuchs series
around the regular singularity at the origin specifying
generalised Bessel and Neumann functions, comple-
mented (ii) in the present paper by asymptotic expan-
sions near the irregular singularity at infinity spec-
ifying generalised Hankel functions. Possible fu-
ture research directions include: (iii) relating gen-
eralised Hankel functions to generalised Bessel and
Neumann functions through confluent hypergeomet-
ric functions; (iv) solving the generalised Bessel dif-
ferential equation using the Laplace transform in the
complex plane to obtain integral representations for
the generalised Bessel, Neumann and Hankel func-
tions; (v) obtaining representations of the latter as
differintegrations of elementary functions using the
Riemann-Liouville fractional derivatives in the com-
plex plane.
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Contribution of Individual Authors to the
Creation of a Scientific Article (Ghostwriting
Policy)
The authors equally contributed in the present re-
search, at all stages from the formulation of the prob-
lem to the final findings and solution.
Sources of Funding for Research Presented in a
Scientific Article or Scientific Article Itself
This work was supported by the Fundação para a
Ciência e Tecnologia (FCT), Portugal, through In-
stitute of Mechanical Engineering (IDMEC), under
the Associated Laboratory for Energy, Transports
and Aeronautics (LAETA), whose grant numbers
are SFRH/BD/143828/2019 to M. J. S. Silva and
UIDB/50022/2020.
Conflicts of Interest
The authors have no conflicts of interest to
declare that are relevant to the content of this
article.
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