Recently, J. A. Tirao [Proc. Nat. Acad. Sci. 100 (14) (2003), 8138–8141]
considered a matrix-valued analogue of the 2F1 Gauß hypergeometric function and
showed that it is the unique solution of a matrix-valued hypergeometric equation
analytic at z = 0 with value I, the identity matrix, at z = 0. We give an independent
proof of Tirao’s result, extended to the slightly more general setting of
hypergeometric functions over an abstract unital Banach algebra. We provide a
similar (but more complicated-looking) result for a second type of noncommutative
2F1 Gauß hypergeometric function.CMUC; FWF Austrian Science Fund grant P17563–N13; FWF Austrian Science Fund grant P17563–N13; FWF, S960

Conflitti, Alessandro

Schlosser, Michael J.

https://thekeep.eiu.edu/commencement_spring2013/1090/thumbnail.jp

Cruse, Beverly J.

The Keep

Ms. Kaci L. Abolt, Student Body President

Eastern Illinois University

https://thekeep.eiu.edu/commencement_spring2013/1051/thumbnail.jp

Ms. Kaci L. Abolt, Student Body President, Mr. Robert K. Martin, Vice President for University Advancement

Estudo Geral

Pre´-Publicac¸o˜es do Departamento de Matema´tica
Universidade de Coimbra
Preprint Number 07–01
ON THE NONCOMMUTATIVE HYPERGEOMETRIC
EQUATION
ALESSANDRO CONFLITTI AND MICHAEL J. SCHLOSSER
Abstract: Recently, J. A. Tirao [Proc. Nat. Acad. Sci. 100 (14) (2003), 8138–8141]
considered a matrix-valued analogue of the 2F1 Gauß hypergeometric function and
showed that it is the unique solution of a matrix-valued hypergeometric equation
analytic at z = 0 with value I, the identity matrix, at z = 0. We give an inde-
pendent proof of Tirao’s result, extended to the slightly more general setting of
hypergeometric functions over an abstract unital Banach algebra. We provide a
similar (but more complicated-looking) result for a second type of noncommutative
2F1 Gauß hypergeometric function.
Keywords: Noncommutative hypergeometric series, hypergeometric equation.
AMS Subject Classification (2000): Primary 33C05; Secondary 33C99, 34G10,
34K30, 46H99.
1. Introduction
Hypergeometric series with noncommutative parameters and argument, in
the special case involving square matrices, have been the subject of recent
study, see e.g. [3, 5, 6, 9, 10, 11, 14, 15, 16]. (For the classical theory of hyper-
geometric series, cf. [2, 7, 8].) In particular, Tirao [16] considered a specific
type of matrix-valued hypergeometric function 2F1, and showed, among oth-
ers results, that it satisfies a matrix-valued differential equation of the second
order (a “matrix-valued hypergeometric equation”), and conversely that any
solution of the latter is a matrix-valued hypergeometric function of the con-
sidered type. This result was reformulated by one of the present authors [15]
in the more general setting of hypergeometric functions with parameters and
argument over an unital Banach algebra R. Specifically, in [14, 15] two re-
lated types of noncommutative hypergeometric series were studied, “type
I” and “type II”, from the view-point of explicit summation theorems they
Received January 8, 2007.
The first author was partly supported by FWF Austrian Science Fund grant P17563–N13, and
partly by CMUC.
The second author was partly supported by FWF Austrian Science Fund grant P17563–N13,
and also partly supported by FWF grant S9607, which is part of the Austrian National Research
Network “Analytic Combinatorics and Probabilistic Number Theory”.
1
2 A. CONFLITTI AND M. J. SCHLOSSER
satisfy. In the terminology of [14, 15], Tirao’s extension of the Gauß hyper-
geometric function belongs to type I. As a matter of fact, the explicit form of
the noncommutative hypergeometric equation satisfied by the type II Gauß
hypergeometric function has so far not been determined. (A priori, it is not
clear that the type II hypergeometric equation would be of second order or
even have a reasonable compact form.) In this paper we give an indepen-
dent derivation of Tirao’s result for the type I Gauß hypergeometric function
and succeed in providing an analogous (however, more complicated-looking)
result for the type II case.
We refer to [4, 17] and [1, 12, 13] for comprehensive references on Banach
algebras, and on differential equations in Banach spaces.
In the following section, we collect some definitions and notations, taken
almost verbatim from [14, 15]. These are needed in Section 3 for the study
of the type I and type II noncommutative hypergeometric equations.
2. Preliminaries
Let R be a unital Banach algebra, i.e., a ring with a multiplicative iden-
tity with some norm ‖ · ‖. Throughout this paper, the elements of R will
be denoted by capital letters A, B, C, . . .. In general these elements need
not commute with each other; however, we may sometimes specify certain
commutation relations explicitly. We denote the identity by I and the zero el-
ement by O. Whenever a multiplicative inverse element exists for any A ∈ R,
we denote it by A−1. (Since R is a unital ring, we have AA−1 = A−1A = I.)
On the other hand, as we shall implicitly assume that all the expressions
which appear are well defined, whenever we write A−1 we assume its exis-
tence. For instance, in (1) and (2) we assume that Ci + jI is invertible for
all 1 ≤ i ≤ r, 0 ≤ j < k.
An important special case is when R is the ring of n × n square matrices
(our notation is certainly suggestive with respect to this interpretation), or,
more generally, one may view R as a space of some abstract operators.
For any nonnegative integers m and l with m ≥ l − 1 we define the non-
commutative product as follows:
m∏
j=l
Aj =
{
I m = l − 1
AlAl+1 · · ·Am m ≥ l.
In [14, 15] a more general definition was given, which however we will not
need here.
NONCOMMUTATIVE HYPERGEOMETRIC EQUATION 3
For nonnegative integers k and r we define the generalized noncommutative
shifted factorial of type I by⌈
A1, A2, . . . , Ar
C1, C2, . . . , Cr
;Z
⌋
k
:=
k∏
j=1
[(
r∏
i=1
(Ci + (k − j)I)
−1(Ai + (k − j)I)
)
Z
]
,
(1)
and the noncommutative shifted factorial of type II by⌊
A1, A2, . . . , Ar
C1, C2, . . . , Cr
;Z
⌉
k
:=
k∏
j=1
[(
r∏
i=1
(Ci + (j − 1)I)
−1(Ai + (j − 1)I)
)
Z
]
.
(2)
Note the unusual usage of brackets (“floors” and “ceilings” are intermixed)
on the left-hand sides of (1) and (2) which is intended to suggest that the
products involve noncommuting factors in a prescribed order. In both cases,
the product, read from left to right, starts with a denominator factor. The
brackets in the form “⌈−⌋” are intended to denote that the factors are falling,
while in “⌊−⌉” that they are rising.
We define the noncommutative hypergeometric series of type I by
r+1Fr
⌈
A1, A2, . . . , Ar+1
C1, C2, . . . , Cr
;Z
⌋
:=
∑
k≥0
⌈
A1, A2, . . . , Ar+1
C1, C2, . . . , Cr, I
;Z
⌋
k
,
and the noncommutative hypergeometric series of type II by
r+1Fr
⌊
A1, A2, . . . , Ar+1
C1, C2, . . . , Cr
;Z
⌉
:=
∑
k≥0
⌊
A1, A2, . . . , Ar+1
C1, C2, . . . , Cr, I
;Z
⌉
k
.
In each case, the series terminates if one of the upper parameters Ai is of the
form −nI. If the series is nonterminating, then the series converges in R if
‖Z‖ < 1. If ‖Z‖ = 1 the series may converge in R for some particular choice
of upper and lower parameters. Exact conditions depend on the Banach
algebra R.
It is clear that given a valid identity of elements in R, one may obtain a
new one by simply reversing all the products (of elements of the unit ring
R) simultaneously on each side of the respective identities. This is clearly
an involution. (For square matrices this is equivalent to transposition of
matrices.) This operation on an expression E ∈ R shall be denoted by ∼E.
4 A. CONFLITTI AND M. J. SCHLOSSER
For instance (compare with (1)),
∼⌈
A1, A2, . . . , Ar
C1, C2, . . . , Cr
;Z
⌋
k
=
k∏
j=1
(
Z
r∏
i=1
(Ai + (j − 1)I)(Ci + (j − 1)I)
−1
)
.
3. Type I and type II noncommutative hypergeometric
equations
Tirao [16] proved the following result:
Proposition. For a positive integer n, let R = Mn×n(C) be the ring of com-
plex n×n square matrices. Let A, B, C, F0 ∈ R be such that the spectrum of C
contains no negative integers, and let z ∈ C. Then F (z) = 2F1
⌈
A, B
C
; zI
⌋
F0
is the unique solution analytic at z = 0 of the matrix-valued hypergeometric
equation
z(1− z)F ′′(z) + (C − z(1 + A + B))F ′(z)−ABF (z) = 0,
where F (0) = F0.
As was indicated without proof in [15, Remark 2.1] this extends easily to
the following:
Theorem 1. Let R be a unital Banach algebra with norm ‖ · ‖, let
A, B, C, F0 ∈ R such that C + jI is invertible for all nonnegative integers j.
Further let Z be central (i.e., Z ∈ {X ∈ R : XY = Y X, ∀Y ∈ R}) with
‖Z‖ < 1. Then
F (Z) = 2F1
⌈
A, B
C
;Z
⌋
F0 (3)
is the unique solution analytic at Z = O of the noncommutative hypergeo-
metric equation
Z(I − Z)F ′′(Z) + (C − Z(I + A + B))F ′(Z)− ABF (Z) = O, (4)
where F (O) = F0.
We provide an operator proof of Theorem 1. On the contrary, Tirao’s
proof of the above Proposition given in [16] is essentially different. Starting
with the matrix-valued hypergeometric equation it involves the computation
of the coefficients Fk in the analytic series F (z) =
∑
k≥0 Fkz
k by a generic
Ansatz.
NONCOMMUTATIVE HYPERGEOMETRIC EQUATION 5
Proof : First of all, the (right multiple of the) type I noncommutative hyper-
geometric series
2F1
⌈
A, B
C
;Z
⌋
F0
=
[∑
k≥0
(
k∏
j=1
(C + (k − j)I)−1(A + (k − j)I)(B + (k − j)I)
)
Zk
k!
]
F0
is clearly analytic at Z = O and 2F1
⌈
A, B
C
;O
⌋
F0 = F0.
Next we show that 2F1
⌈
A, B
C
;Z
⌋
F0 is a solution of the differential equation
(4). We define the linear operator
DT := T + Z
d
dZ
,
where T ∈ R, acting (from the left) on functions of Z over R.
If F (Z) is analytic at Z = O we can write F (Z) =
∑
k≥0 FkZ
k, where
Fk ∈ R for any nonnegative integer k. It is immediate that
DT F (Z) =
∑
k≥0
(T + kI)FkZ
k.
Hence
DA
(
DB 2F1
⌈
A, B
C
;Z
⌋)
=
∑
k≥0
(A + kI)(B + kI)
⌈
A, B
C, I
;Z
⌋
k
,
and
DC−I 2F1
⌈
A, B
C
;Z
⌋
=
∑
k≥0
(C + (k − 1)I)
⌈
A, B
C, I
;Z
⌋
k
= C − I +
∑
k≥1
(A + (k − 1)I)(B + (k − 1)I)
×
(
k−1∏
j=1
(C + (k − 1− j)I)−1(A + (k − 1− j)I)(B + (k − 1− j)I)
)
Zk
k!
= C − I +
∑
k≥0
(A + kI)(B + kI)
6 A. CONFLITTI AND M. J. SCHLOSSER
×
(
k∏
j=1
(C + (k − j)I)−1(A+ (k − j)I)(B + (k − j)I)
)
Zk+1
(k + 1)!
.
Thus we have
d
dZ
(
DC−I 2F1
⌈
A, B
C
;Z
⌋)
= DA
(
DB 2F1
⌈
A, B
C
;Z
⌋)
. (5)
Since the differential equation
d
dZ
(DC−I F (Z)) = DA (DB F (Z)) , (6)
or, more explicitly,
d
dZ
(
C − I + Z
d
dZ
)
F (Z) =
(
A + Z
d
dZ
)(
B + Z
d
dZ
)
F (Z),
is equivalent to (4), it follows from (5) (and multiplication of a constant from
the right) that 2F1
⌈
A, B
C
;Z
⌋
F0 satisfies the differential equation (4).
The uniqueness of the solution (3) of (4) with F (0) = F0 readily fol-
lows from the theorem of existence and uniqueness of solutions of differential
equations in Banach spaces (hence in Banach algebras), cf. e.g. [13]. All
we need to show is that if there were two solutions F1(Z) and F2(Z) then
F ′1(O) = F
′
2(O). (As we are considering a second order differential equa-
tion, two initial conditions, fixing F (O) and F ′(O), are required to make the
solution unique.)
Asume that F1(Z) and F2(Z) are solutions of (4) with F1 (O) = F2 (O) =
F0. Then we have
Z(I − Z)F ′′1 (Z) + (C − Z(A+ B + I))F
′
1(Z)−ABF1(Z)
= Z(I − Z)F ′′2 (Z) + (C − Z(A+ B + I))F
′
2(Z)−ABF2(Z).
Evaluating this equation in Z = O we get C F ′1(O) = C F
′
2(O) and since C
is invertible the claim follows.
Now we are ready to state and prove the following new result concerning
type II noncommutative hypergeometric series. It appears to lie in the nature
of the type II series that the result is not as simple and elegant as in the
corresponding type I case. In particular, the following theorem as stated
requires the condition C(C − A − B) + AB being invertible, which has no
counterpart in the type I case.
NONCOMMUTATIVE HYPERGEOMETRIC EQUATION 7
Theorem 2. Let R be a unital Banach algebra with norm ‖ · ‖, let
A, B, C, F0 ∈ R such that C(C − A − B) + AB and C + jI are invertible
for all nonnegative integers j. Further let Z be central (i.e., Z ∈ {X ∈ R :
XY = Y X, ∀Y ∈ R}) with ‖Z‖ < 1. Then
F (Z) = F0 2F1
⌊
A, B
C
;Z
⌉
(7)
is the unique solution analytic at Z = O of the noncommutative hypergeo-
metric equation
Z(I−Z)F ′′(Z)+ZF ′(Z)(C−I−A−B)+
(
(I − Z)F ′(Z)− F (Z)C−1AB
)
×
(
C(C − A− B) + AB
)−1
C
(
C(C − A− B) + AB
)
= O, (8)
where F (O) = F0.
Proof : First of all, the (left multiple of the) type II noncommutative hyper-
geometric series
F0 2F1
⌊
A, B
C
;Z
⌉
= F0
∑
k≥0
(
k∏
j=1
(C + (j − 1)I)−1(A + (j − 1)I)(B + (j − 1)I)
)
Zk
k!
is clearly analytic at Z = O and F0 2F1
⌊
A, B
C
;O
⌉
= F0.
Next we show that F0 2F1
⌊
A, B
C
;O
⌉
is a solution of the differential equation
(8). We have
d
dZ
2F1
⌊
A, B
C
;Z
⌉
=
∑
k≥1
(
k∏
j=1
(C + (j − 1)I)−1(A+ (j − 1)I)(B + (j − 1)I)
)
Zk−1
(k − 1)!
=
∑
k≥0
(
k+1∏
j=1
(C + (j − 1)I)−1(A+ (j − 1)I)(B + (j − 1)I)
)
Zk
k!
8 A. CONFLITTI AND M. J. SCHLOSSER
=
∑
k≥0
⌊
A, B
C, I
;Z
⌉
k
(C + kI)−1(A + kI)(B + kI). (9)
We define the linear operator D˜T by
D˜T := T +
d˜
dZ
Z,
where T ∈ R, acting from the right on functions over R. Here d˜
dZ
is the
differential operator applied from the right side. With other words
F (Z)
d˜
dZ
=
d
dZ
F (Z),
and
F (Z) D˜T = F (Z)T + Z
d
dZ
F (Z),
where F (Z) is any function of Z (Z being central) over R.
In particular, we have
F (Z) D˜T =
∼(DT (
∼F (Z))) ,
where the reversion operator ∼ was defined at the end of Section 2.
If F (Z) is analytic at Z = O we can write F (Z) =
∑
k≥0 FkZ
k, where
Fk ∈ R for any nonnegative integer k. It is immediate that
F (Z) D˜T =
∑
k≥0
FkZ
k (T + kI),
and
F (Z) D˜−1U =
∑
k≥0
FkZ
k (U + kI)−1,
provided U + kI is invertible in R for all nonnegative integers k.
Hence ((
2F1
⌊
A, B
C
;Z
⌉
D˜−1C
)
D˜A
)
D˜B
=
∑
k≥0
⌊
A, B
C, I
;Z
⌉
k
(C + kI)−1(A+ kI)(B + kI)
NONCOMMUTATIVE HYPERGEOMETRIC EQUATION 9
= 2F1
⌊
A, B
C
;Z
⌉
d˜
dZ
,
by (9).
It follows that
G(Z) = 2F1
⌊
A, B
C
;Z
⌉
D˜−1C
is a solution of the differential equation(
G(Z) D˜A
)
D˜B =
(
G(Z) D˜C
) d˜
dZ
.
This is simply a “reversed” version of (6) with A and B interchanged and
C + I in place of C. It thus follows from Theorem 1 that G(Z) satisfies the
reversed type I noncommutative hypergeometric equation:
Z(I − Z)G′′(Z) + G′(Z)(C + I − Z(I + A + B))−G(Z)AB = O. (10)
We now need to rewrite (10) in terms of F (Z) = 2F1
⌊
A, B
C
;Z
⌉
. We have
F (Z) = G(Z) D˜C = G(Z)C + ZG
′(Z),
and
F ′(Z) = G′(Z)(C + I) + ZG′′(Z),
which, in conjunction with (10), gives
(I−Z)F ′(Z)+F (Z)(C−A−B)−F (Z) D˜−1C
(
C(C−A−B)+AB
)
= O. (11)
Next, we multiply both sides of (11) from the right with(
C(C −A −B) + AB
)−1
D˜C
(
C(C −A −B) + AB
)
(which is
(
C(C −A−B) +AB
)−1
C
(
C(C −A−B) +AB
)
+ d˜
dZ
Z). After a
series of computations, including the simplifiction
I−(C−A−B)
(
C(C−A−B)+AB
)−1
C = C−1AB
(
C(C−A−B)+AB
)−1
C,
we eventually arrive at (8). Further, by multiplying a constant from the left,
it follows that F0 2F1
⌊
A, B
C
;Z
⌉
satisfies the differential equation (8).
Using the same argument as in the proof of Theorem 1, one readily estab-
lishes the uniqueness of the solution (7) of (8) with F (O) = F0.
10 A. CONFLITTI AND M. J. SCHLOSSER
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Alessandro Conflitti
CMUC, Centre for Mathematics, University of Coimbra, Apartado 3008, 3001-454 Coim-
bra, Portugal
E-mail address : conflitt@mat.uc.pt
URL: http://www.mat.uc.pt/~conflitt
Michael J. Schlosser
Fakulta¨t fu¨r Mathematik, Universita¨t Wien, Nordbergstraße 15, A-1090 Vienna, Aus-
tria
NONCOMMUTATIVE HYPERGEOMETRIC EQUATION 11
E-mail address : michael.schlosser@univie.ac.at
URL: http://www.mat.univie.ac.at/~schlosse
