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Electronic Transactions on Numerical Analysis.
Volume 24, pp. 94-102, 2006.
Copyright  2006, Kent State University.
ISSN 1068-9613.
Kent State University
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NONCOMMUTATIVE EXTENSIONS OF
RAMANUJAN’S SUMMATION MICHAEL SCHLOSSER Abstract. Using functional equations, we derive noncommutative extensions of Ramanujan’s summation.
Key words. noncommutative basic hypergeometric series, Ramanujan’s summation
AMS subject classifications. 33D15, 33D99
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. the papers by Duval and Ovsienko [4], Grünbaum [6], Tirao [16], and some of the
references mentioned therein. Of course, this subject is also closely related to the theory
of orthogonal matrix polynomials which was initiated by Krein [12] and has experienced a
steady development, see e.g. Durán and López-Rodrı́guez [3].
Very recently, Tirao [16] considered a particular type of a matrix valued hypergeometric
function (which, in our terminology, belongs to noncommutative hypergeometric series of
“type I”). He showed, in particular, that the matrix valued hypergeometric function satisfies
the matrix valued hypergeometric differential equation, and conversely that any solution of
the latter is a matrix valued hypergeometric function.
In [14], the present author investigated hypergeometric and basic hypergeometric series
involving noncommutative parameters and argument (short: noncommutative hypergeometric series, and noncommutative basic or -hypergeometric series) over a unital ring (or,
when considering nonterminating series, over a unital Banach algebra ) from a different,
nevertheless completely elementary, point of view. These investigations were exclusively devoted to the derivation of summation formulae (which quite surprisingly even exist in the
noncommutative case), aiming to build up a theory of explicit identities analogous to the rich
theory of identities for hypergeometric and basic hypergeometric series in the classical, commutative case (cf. [15] and [5]). Two closely related types of noncommmutative series, of
“type I” and “type II”, were considered in [14]. Most of the summations obtained there concern terminating series and were proved by induction. An exception are the noncommutative
extensions of the nonterminating -binomial theorem [14, Th. 7.2] which were established
using functional equations. Aside from the latter and some conjectured -Gauß summations,
no other explicit summations for nonterminating noncommutative basic hypergeometric series were given. Furthermore, noncommutative bilateral basic hypergeometric series were
not even considered.
In this paper, we define noncommutative bilateral basic hypergeometric series of type I
and type II (over an abstract unital Banach algebra ) and prove, using functional equations,
noncommutative extensions of Ramanujan’s summation. These generalize the noncommutative -binomial theorem of [14, Th. 7.2]. Our proof of the sum here is similar to
Andrews and Askey’s [1] proof in the classical commutative case. Ramanujan’s summation (displayed in (2.4)) is one of the fundamental identities in -series. It is thus just natural
to look for different extensions, including noncommutative ones.
This paper is organized as follows. In Section 2, we review some standard notations for
Received December 3, 2004. Accepted for publication June 7, 2005. Recommended by J. Arvesú.
Fakultät für Mathematik, Universität Wien, Nordbergstraße 15, A-1090 Wien, Austria
(michael.schlosser@univie.ac.at), http://www.mat.univie.ac.at/˜schlosse.
94
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95
NONCOMMUTATIVE SUMMATIONS
basic hypergeometric series and then explain the notation we utilize in the noncommutative
case. Section 3, is devoted to the derivation of noncommutative summations.
We stress again, as in [14], that by “noncommutative” we do not mean “ -commutative”
or “quasi-commutative” (i.e., where the variables satisfy a relation like ; such series
are considered e.g. in [11] and [17]) but that the parameters in our series are elements of some
noncommutative unital ring (or unital Banach algebra).
2. Preliminaries.
2.1. Classical (commutative) basic hypergeometric series. For convenience, we recall some standard notations for basic hypergeometric series (cf. [5]). When considering the
noncommutative extensions in Subsection 2.2 and in Section 3, the reader may find it useful
to compare with the classical, commutative case.
Let be a complex number such that . Define the -shifted factorial for all
integers (including infinity) by
"!
%
*
$#&%(')
.-
+&,
+
#0/
We write
(2.1)
:
98
8 /3/;/ 8 1
132"154 76 <
:
<
<
8
8 /33
1 4
/ / 8 3
!
D !
D E:F!
!
B
1 $# %
$# %
$# % /3/3/
= % 8
< !
<
!
')A@
8
5
1
4
$
&
#
%
$
G
#
H
%
3
/
;
/
/
$
&
#
%
,
C
%
8>=$?
to denote the (unilateral) basic hypergeometric 1 2 154
(2.2)
1 1 6 <
8
:
8 ;
/ /3/ 8 1
<8 : 8
<8
1
/3/3/
!
8&=F?
B
'I
series. Further, we write
D
!
<
@
% , 4
$#&%J/3/3/
!
$# % /3/3/
<
1
1
!
!
$#G% = % 8
$# %
@
1 1 series.
to denote the bilateral basic
hypergeometric
<
<
8
8
1
In (2.1) and (2.2),
are called the upper parameters, 8 /3/3/ 8 1 the lower pa/;/3/
1 1 series in (2.2)
rameters, = is the argument, and the base of the series. The bilateral
<
1
;
1
"
2
3
1
4
reduces to a unilateral
series if one of the lower parameters, say , equals (or more
generally, an integral power of ).
The basic hypergeometric 1;2K134 series terminates if one of the upper parameters, say
4ML
1 , is of the form , for a nonnegative integer N . If the basic hypergeometric series does
= OP . Similarly, the bilateral basic
not terminate then it converges by the ratio test when
<
<
=
hypergeometric series converges when Q and /;/3/ 1;R
/;/3/ 1 = ES .
We recall two important summations. One of them is the (nonterminating) -binomial
theorem,
2 C 6
(2.3)
!
-
8&=F?
=
=
!
$#
!
8
$# @
@
where = "T (cf. [5, Appendix (II.3)]). It was discovered independently by several mathematicians, including Cauchy, Gauß, and Heine. A bilateral extension of (2.3) was found by
the legendary Indian mathematician Ramanujan (see Hardy [8]),
(2.4)
76 <
<
!
8&=F?
!
$#
< !
$# @
<
@
R
R
"!
"!
$#
$# @
@
=
=
!
!
$#
$# @
@
<
!
R = $#
!
R = $
# @
@
8
where = OU and R = U (cf. [5, Appendix (II.29)]). Unfortunately, Ramanujan (who
very rarely gave any proofs) did not provide a proof of the above bilateral summation. The
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96
MICHAEL SCHLOSSER
first proof of (2.4) was given by Hahn [7, VW in Eq. (4.7)]. Other proofs were given by
Jackson [10], Ismail [9], Andrews and Askey [1], the author [13, Sec. 3], and others. Some
immediate applications of Ramanujan’s summation formula to arithmetic number theory are
considered in [2, Sec. 10.6].
2.2. Noncommutative basic hypergeometric series. Most of the following definitions
are taken from [14]. However, the definitions for noncommutative bilateral basic hypergeometric series in (2.8) and (2.9) (although obvious) are new.
Let be a unital ring (i.e., a ring with a multiplicative identity). When considering
infinite series and infinite products of elements of we shall further assume that is a
Banach
algebra (with some norm XHYEX ). The elements of will be denoted by capital letters
Z
8>[\8 /3/3/ . In general these elements do not commute with each other; however, we may
sometimes specify certain commutation relations explicitly. We denote the identity
by ] and
ZT_
the zero element
by
^ . Whenever a multiplicative inverse element exists for any
, we
Z 4
Z`Z 4
Z 4 Z
denote it by
. (Since is a unital ring, we have
a] .) On the other
hand, as we shall implicitly
assume that all the expressions which appear are well defined,
Z 4
whenever we + write
we assume its existence. For instance, in (2.6) and (2.7) we assume
that ]b- [dc is invertible for all fehgehi , jeWklm .
An important special case is when is the ring of NonjN square matrices (our notation is
certainly suggestive with respect to this interpretation), or, more generally, one may view as a space of some abstract operators. _
Let p be the set of integers. For q 8Gr
ptsvu$wyxmz we define the noncommutative product
as follows:
{
{
Z | Z ‚| 
Z
{
/;/3/
Z 4
Z 4 :
Z 4

|4
|4
/;/3/
€
Z +
~}
*
+&,|
r
qM-h
r„ƒ q
r
/
qM-h
{
Note that
*
(2.5)
+&,|
Z +
|4
*{
{
Z 4
+&,

| 4 + 8
_
for all q 8&r
p(s†u$wyxmz .
Throughout this paper, will be a parameter which commutes with any of the other
parameters appearing in the series. (For instance,
a central element such as Q9] , a scalar
_
multiple of_ the unit element in , for 9]
, trivially satisfies this requirement.)
Let pds\u9x‡z . We define the generalized noncommutative -shifted factorial of type
I by ˆ
:
8 Z
8 /;/3/ 8 Z 1
:
[ 98&[
f
8 3
/ /3/ 8&[ 1
Z
(2.6)
!
80‰‹Š
%
'I
%
*
1
*
+&, fŒ c ,
]b-
+
[ c % 4 # 4
]b-
Z c % 4 +
#Ž
‰H
/
Similarly, we define the generalized noncommutative -shifted factorial of type II by
 Z
(2.7)
[
:
Z
8 Z
8 ;
/ /3/ 8 1
:
>8 [
8 ;
/ /3/ 8>[ 1
!
8‰.‘
%
'I
*
%
*
+&, bŒ c ,
1
]y-
[ c + 4
#
4
]b-
Z c + 4
#’Ž
‰‹ /
Note the unusual usage of brackets (“floors” and “ceilings” are intermixed) on the lefthand sides of (2.6) and (2.7) 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
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97
NONCOMMUTATIVE SUMMATIONS
with a denominator factor. The brackets in the form “ “-‹” ” are intended to denote that the
factors are falling, while in “ •-‹– ” that they are rising.
ˆ
We defineˆ the noncommutative basic hypergeometric
series of type I by
Z :
Z
$8
8 /;/3/ 8 1
:
[ 8>[
y
8 /3;
/ / 8>[ 134
Z
1 2 154
!
8‰.Š
Z
B
')
Z :
Z
8 3
/ /;/ 8 1
8 /3/;/ 8>[ 134 98 8
[y8>[
:
% — C
˜
!
80‰‹Š
80‰ ‘
8
%
and the noncommutative basic hypergeometric series of type II by

Z :
Z
8 ;
/ /3/ 8 1
:
8>[
8 /3/;/ 8>[ 134
Z
1˜2"154
8
[
!
8‰ ‘

B
')
Z
[
% — C
˜
8
8>[
Z :
:
Z
8 3
/ /;/ 8 1
8 /3/;/ 8>[ 134 8 !
/
%
ˆ the noncommutative bilateral basic
ˆ hypergeometric series of type I by
Further, we define
:
Z
8 Z
8 ;
/ /3/ 8 1 !
:
8&[
8 ;
/ /3/ 8&[ 1
Z
1 1
(2.8)
[
8‰ Š
'I
:
Z
8 Z
8 ;
/ /3/ 8 1
:
8>[
8 ;
/ /3/ 8>[ 1
Z
B
@
[
% , 4
!
8‰ Š
8
%
@
and the noncommutative bilateral basic hypergeometric series of type II by
 Z
1 1
(2.9)
[
:
8 Z
8 /3/;/ 8 Z 1
:
8&[
8 3
/ /;/ 8&[ 1
!
80‰ ‘
 Z
B
')
@
:
8 Z
8 /3/3/ 8 Z 1 !
:
8&[
8 3
/ /3/ 8&[ 1
[
% , 4
80‰ ‘
/
%
@
We also refer to the respective series as (noncommutative) -hypergeometric series. In
each
case (of type I and type II), the 1;2"134 series terminates if one of the upper parameters
Z c
4ML
is of the form . If the 1˜2"154 series does not terminate, then (implicitly assuming that
is a unital Banach algebra with some norm X™YX ) it œ
converges
when X ‰ XšP . Similarly,
1
Z 4
4
‰
‰

1
1
,
c
1
4 c [ 1  4 c XvQ .
the
series converges in when X Xv› and X
We also consider reversed (or “transposed”) versions of generalized noncommutative shifted factorials and noncommutative bilateral basic hypergeometric series of type I and II.
These are ˆdefined as follows (compare with (2.6), (2.7), (2.8) and (2.9)):
:
Z
8 Z
8 3
/ /3/ 8 1
:
&8 [
8 3
/ /3/ 8&[ 1
Z

[

 Z
[
:
Z
8 Z 8 3
/ /;/ 8 1
:
>8 [
8 3
/ /;/ 8&[ 1
!
!
80‰‹Š
%
8‰.‘
%
')
')
%
*
+&, v
*
c,
%
1
*
‰
+&, v
1
*
‰
c,
ˆ
Z c + 4
]b-
Z c % 4 + # ]b-
# b
] -
[ c + 4
#
4
8
+
[ c % 4 # 4
Ž
Ž
8
ˆ
Z

1 1
]y-
[
:
8 Z
8 /;/3/ 8 Z 1 !
:
8&[
8 ;
/ /3/ 8&[ 1
8‰.Š
'I
:
Z
8 Z
8 3
/ /3/ 8 1
:
&8 [
8 3
/ /;/ 8&[ 1
!
 Z
:
Z
$8 Z
8 3
/ /3/ 8 1
:
[f$8&[
8 3
/ /;/ 8&[ 1
!
Z

B
@
% , 4
[
80‰‹Š
80‰ ‘
8
%
@
and
1 1

 Z
:
Z
$8 Z
8 ;
/ /3/ 8 1 !
:
[f98&[
8 ;
/ /3/ 8&[ 1
8‰ ‘
'I
B

@
% , 4
@
%
/
Of course, reversed versions of the unilateral noncommutative basic hypergeometric series
are defined analogously.
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MICHAEL SCHLOSSER
3. Noncommutative summations. In [14, Th. 7.2], the following two noncommutative extensions of the nonterminating -binomial theorem (which generalize [5, II.3]) were
given.
Z
P ROPOSITION 3.1. Let and ‰ be noncommutative
parameters of some unital Banach
Z
algebra, and suppose that commutes with both and ‰ . Further, assume that X ‰ Xšž .
Then we have the following summation for a noncommutative basic hypergeometric series of
ˆ
type I.
Z
2 C
(3.1)
!
-
 Z
8‰ Š
‰
!
‰
8 ] ‘
/
@
Further, we we have the following summation for a noncommutative basic hypergeometric
series of type II.
 Z
2 C
(3.2)
!
-
80‰ ‘

 Z
‰
‰
!
8 ] ‘
/
@
Here we extend Proposition 3.1 to summations for bilateral series. Our proof is similar
to that of the classical result given in [1] (see also [2, p. 502, first proof of Th. 10.5.1]), but
also similar to the proof of
Proposition 3.1 given in [14].
Z
T HEOREM 3.2. Let , [ and ‰ be noncommutative parameters of some unital Banach
algebra, suppose that and [ both
commute with any of the other parameters. Further,
4 Z 4
Xjž . Then we have the following summation for a
assume that X ‰ XšT and X [j‰
ˆ
noncommutative bilateral
basic hypergeometric series of type I.
(3.3)
Z
0

!
[
8‰.Š
ˆ
&8 [j‰ 4 Z 4 ‰ !
[\8&[j‰ 4 Z 4
‰ 4 Z 4 ‰ 4 Z 4 ‰ 8 ] ‘
!
 Z
‰ !
‰
8 ] Š
@
8 ] ‘
@
/
@
Further, we have the following summation for a noncommutative bilateral basic hypergeometric series of type II.
(3.4)
0

 Z
!
[
 Z
‰
‰
!
8‰.‘

8 ] ‘
‰ 4
Z 4
[\8 Z 4
8 @
!

8 ] ‘

[ Z 4
[j‰ 4 Z 4
!
@
8 ] ‘
/
@
Clearly, Theorem 3.2 reduces to Proposition 3.1 when [ Q
.
Proof of Theorem 3.2. We prove (3.3) using (3.1), leaving the proof of (3.4) using (3.2),
which is similar,
to the reader.
Z
8>[\8‰ # denote the series on the left-hand side of (3.3). We make use of the two
Let Ÿ
simple identities
‰
(3.5a)
(3.5b)
](
Z ‰
Z
% .
% ‰™8
]¡
#
% 8
[ %. [
]b
#
to obtain two functional equations for Ÿ . We also make use of the simple relation
(3.6)
Ÿ
Z
8&[¢8‰ #£Ÿ Z 8>[ 80‰ # y
] -
[ # 4
]b-
Z
# ‰.8
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99
NONCOMMUTATIVE SUMMATIONS
obtained by shifting the summation index in Ÿ by one.
First, (3.5a) gives
Z
‰ Ÿ
(3.7)
Z
8&[¢8‰ #t
Z
‰ Ÿ
8&[¢8‰ b# Z
Ÿ
8&[\80‰ # y
] Z
# ‰.8
while (3.5b) gives
(3.8)
Z
Ÿ
8&[ 8‰ #7
[ Ÿ
Z
80‰ b# 8>[ ]¡-
[
4
Zy4
Z
#GŸ
8&[¢8‰ #0/
Combining (3.8), (3.7), and (3.6), one readily deduces
Z
Ÿ
8&[
Z
[ G# Ÿ
8>[\8‰
Z
4 Z 4
- [š‰
Ÿ
Z
8>[\8‰
]b- [ #GŸ
Z
4 Z 4
- [š‰
Ÿ
8‰ #£
]b-
[š‰
Z
8>[ 80‰ #
8&[ 8 ‰ # b
] # ‰
Z
4
4
[š‰
‰ Ÿ Z >8 [ #
8>[\80‰ # ]b- [ # 8
80‰ #
#
‰ Ÿ
Z
or equivalently
4
[š‰
]b-
Z 4
Z
8&[
‰ #GŸ
8‰ #£
]b-
[ # b
] -
4
[š‰
Z 4
Z
8&[¢8‰ # 8
#GŸ
thus
Z
8&[\80‰ #£
(3.9) Ÿ
[j‰ 4
]b-
Zy4
#
4
[š‰ 4
]b-
‰ # ¡
] -
Zy4
[
4
#
Z
8&[
Ÿ
8‰ #0/
Iteration of (3.9) gives
Z
8&[¢8‰ #£
(3.10) Ÿ
*
@
] ¡
&+ ,Cv¤
[j‰ 4
Zy4
+
4
#
]b-
[š‰ 4
Zy4
‰ +
# b
] -
[ +
4 ¥
#
Z
8 ^
Ÿ
80‰ #5/
Z
8 ^ 80‰ # . It is not easy to do this directly but we know the value
We still
need to compute Ÿ
Z 8
of Ÿ
8‰ # (by Proposition 3.1). We set [ ›
in (3.10) which gives
Z
8 Ÿ
8‰ #7
*
@
] ¡
&+ ,C`¤
‰ 4
Zb4
+&
#
4
]¡-
‰ 4
thus we obtain
(3.11)
Ÿ
Z
8 ^
Zy4
‰ +&
+&
# ¡
] -¦
4 >¥
#
Ÿ
Z
8 ^
80‰ # 8
ˆ
8‰ #£
*
+&
6 @
]y-¦
+&,C
# ?
‰ 4 Z 4 ‰ 4 Z 4 ‰ !
8 ] Š
@
Ÿ
Z
8 8‰ #0/
Combination of (3.10), (3.11) and (3.1) establishes the result.
Z
In the Z summations of Theorem 3.2, the lower parameter [ commutesZ with both
and ‰ while doesZ not commute with ‰ . In the next theorem the roles of and [ are
interchanged. Here commutes with both [ and ‰ while [ does not commute with ‰ .
Z
T HEOREM 3.3. Let Z , [ and ‰ be noncommutative parameters of some Banach algebra, suppose that and both commute with any of the other parameters. Further, assume
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100
MICHAEL SCHLOSSER
Z 4
4
XyS . Then we have the following summation for a noncomthat X ‰ Xd§ and X [š‰
ˆ
mutative bilateral
basic hypergeometric series of type I.
Z
(3.12) 0
‰
!
[

4
8‰ Š
ˆ
8‰`[š‰ 4 Z 4
Z 4
8 ‰
0
!
Z ‰
‰™[j‰ 4
8 ] ‘

!
‰ 4 Z 4 [š‰ 4 Z 4
8 ] Š
@
!
8 ] ‘
‰ /
@
@
Further, we have the following summation for a noncommutative bilateral basic hypergeometric series of type II.
 Z
0£
(3.13)
!
[


‰ 4
8‰.‘
‰ 4 Z 4 [j‰ 4 Z 4

!
Z ‰™8
Z 4
8>[
8 ] ‘
!
 Z 4

8 ] ‘
[
!
‰
@
8 ] ‘
@
‰ /
@
Proof. We indicate the derivation of (3.12) from (3.3). (The derivation of (3.13) from
(3.4) is analogous.) The sum on the left-hand side of (3.3) remains unchanged if the summation index,
ˆ say , is replaced by -` . Using (2.5), we compute
Z
!
[
8‰.Š
4 %
*
4 %
+&,
¤
*
C
4 % ¤
+&,
%
*
¤
+&,
%
*
+&,
¤
‰ 4
‰
4
Z 4
ˆ
Zb4
Z
]¡-
[
#
Z 4
!
Z
]¡
4
4 % "+
]y-
4
#
4
Z 4
4
M+
4
[ ]b-
4
‰
4 % 4 +
[ ]y-
4 % 4 +
Z
]¡
4
#
8>[š‰ 4
Zb4
4
#
Š
%
Z©¨ª
¥ 4
# ‰
4 % "+ ¥
#
4
[ ]b-
 % 4 +
"+
4
¥
# ‰
[
]b-
4
 % 4 +
¥
# [
Z ‰
/
¨ª
4
Z 4
¨ª
[
, [
, ‰
Thus, by performing the simultaneous replacements
[š‰ 4 Z 4 , in (3.3), we obtain (3.12).
We complete this paper with four more 5 summations, immediately obtained from
corresponding summations in Theorems 3.2 and 3.3 by “reversing all products” (cf. [14, Subsec. 8.2]) on each side of the respective identities.
Z
T HEOREM 3.4. Let , [ and ‰ be noncommutative parameters of some unital Banach
algebra, suppose that andZ [ both commute with any of the other parameters. Further,
4 ‰ 4 [
assume that X ‰ XšT and X
Xjž . Then we have the following summation for a
ˆ
reversed noncommutative
bilateral basic hypergeometric series of type I.

Z

0

!
[
‰ Z
‰
!
8‰.Š
ˆ

8 ] ‘
@
Z 4 ‰ 4
‰ Z 4 ‰ 4 !

8 ] Š
@

‰ Z 4 ‰ 4 [¢8 Z 4 ‰ 4 [\8>[
!
8 ] ‘
@
/
ETNA
Kent State University
etna@mcs.kent.edu
101
NONCOMMUTATIVE SUMMATIONS
Further, we have the following summation for a noncommutative bilateral basic hypergeometric series of type II.
 Z

0
!
[

Z 4
Z 4
8‰.‘
‰ 4
[
!
[

8 ] ‘

8
Z 4
Z 4
‰ 4
8>[
!

!
‰
@
Z
Z
‰
8 ] ‘
8 ] ‘
/
@
@
T HEOREM 3.5. Let Z , [ and ‰ be noncommutative parameters of some Banach algebra, suppose that Z and both commute with any of the other parameters. Further, assume
4 ‰ 4 [
Xf« . Then we have the following summation for a noncomthat X ‰ Xd« and X
ˆ
mutative bilateral
basic hypergeometric series of type I.
‰
Z

5
!
[
 Z 4
Z 4

80‰‹Š
‰ 4
‰ 4
ˆ
[
!

8 ] ‘
‰ 4
‰ Z
!
[š‰

8 ] Š
@
 Z 4
‰ 4 š
[ ‰™8 ‰™8 Z 4 !
8 ] ‘
@
‰
4
/
@
Further, we have the following summation for a noncommutative bilateral basic hypergeometric series of type II.

[

‰
 Z
[ Z 4
‰
!
80‰ ‘
!

8 ] ‘
@

8 ‰ Z
0
Z
[ 8 4 \
!
 Z 4
Z 4
8 ] ‘
@
‰ 4
‰ 4
[
!
8 ] ‘
‰
4
/
@
Acknowledgements. Supported by Austrian Science Fund FWF, grant P17563-N13,
and partly by EC’s IHRP Programme, grant HPRN-CT-2001-00272 “Algebraic Combinatorics in Europe”.
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ETNA
Kent State University
etna@mcs.kent.edu
102
MICHAEL SCHLOSSER
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