RAMANUJAN’S MASTER THEOREM
TEWODROS AMDEBERHAN, OLIVIER ESPINOSA, IVAN GONZALEZ,
MARSHALL HARRISON, VICTOR H. MOLL, AND ARMIN STRAUB
Abstract. S. Ramanujan introduced a technique, known as Ramanujan’s Master Theorem, which provides an explicit expression for the
Mellin transform of a function in terms of the analytic continuation of
its Taylor coefficients. The history and proof of this result are reviewed,
and a variety of applications is presented. Finally, a multi-dimensional
extension of Ramanujan’s Master Theorem is discussed.
1. Introduction
Ramanujan’s Master Theorem refers to the formal identity
Z ∞
x
x2
s−1
(1.1)
x
λ(0) − λ(1) + λ(2) − · · · dx = Γ(s)λ(−s)
1!
2!
0
stated by S. Ramanujan’s in his Quarterly Reports [2, p. 298]. It was widely
used by him as a tool in computing definite integrals and infinite series. In
fact, as G. H. Hardy puts it in [14], he “was particularly fond of them [(1.1)
and (2.4)], and used them as one of his commonest tools.”
The goal of this semi-expository paper is to discuss the history of (1.1)
and to describe a selection of applications of this technique. Section 2 discusses evidence that (1.1) was nearly discovered as early as 1874 by J. W.
L. Glaisher and J. O’Kinealy. Section 3 briefly outlines Hardy’s proof of
Ramanujan’s Master Theorem. The critical issue is the extension of the
function λ from N to C. Section 4 presents the evaluation of a collection of
definite integrals with most of the examples coming from the classical table
[13]. Further examples of definite integrals are given in Section 8 which
collects integrals derived from classical polynomials.
Section 5 is a recollection on the evaluation of the quartic integral
Z ∞
dx
(1.2)
N0,4 (a; m) =
.
4 + 2ax2 + 1)m+1
(x
0
This section provides a personal historical context: it was the evaluation of
(1.2) that lead one of the authors to (1.1).
Date: July 10, 2011.
1991 Mathematics Subject Classification. Primary 33B15, Secondary 33C05, 33C45.
Key words and phrases. integrals, analytic continuation, series representation, hypergeometric functions, random walk integrals, method of brackets.
1
2
RAMANUJAN’S MASTER THEOREM
Sections 6 and 9 outline the use of Ramanujan’s Master Theorem to ongoing research projects: Section 6 deals with an integral related to the distance
traveled by a uniform random walk in a fixed number of steps; finally, Section 9 presents a multi-dimensional version of the main theorem that has
appeared in the context of Feynman diagrams.
The use of Ramanujan’s Master Theorem has been restricted here mostly
to the evaluation of definite integrals. Many other applications appear in the
literature. For instance, Ramanujan himself employed it to derive various
expansions: the two examples given in [14, 11.9] are the expansion of e−ax
in powers of xebx as well as an expansion of the powers xr of a root of
aqxp + xq = 1 in terms of powers of a.
2. History
The first integral theorem in the spirit of Ramanujan’s Master Theorem
appears to have been given by Glaisher in 1874, [10]:
Z ∞
π
(2.1)
a0 − a1 x2 + a2 x4 − . . . dx = a− 1 .
2 2
0
Glaisher writes, “of course, an being only defined for n a positive integer,
a− 1 is without meaning. But in cases where an involves factorials, there is
2
a strong presumption, derived from experience in similar questions, that the
formula will give correct results if the continuity of the terms is preserved
by the substitution of gamma functions for the factorials. This I have found
to be true in every case to which I have applied (2.1).”
Glaisher in [10] formally obtained (2.1) by integrating term-by-term the
identity
x2
x4
a0
2
−∆a
+∆
a
−· · · .
0
0
1 + x2
(1 + x2 )2
(1 + x2 )3
Here ∆ is the forward-difference operator defined by ∆an = an+1 − an .
Glaisher’s argument, published in July 1874, was picked up in October of
the same year by O’Kinealy who critically simplified it in [15]. Employing
the forward-shift operator E defined by E ·λ(n) = λ(n+1), O’Kinealy writes
the left-hand side of (2.2) as 1+x12 E · a0 which he then integrates treating E
as a number to obtain
π −1/2
π
E
· a0 = a− 1 ,
2
2 2
thus arriving at the identity (2.1). O’Kinealy, [15], remarks that “it is
evident that there are numerous theorems of the same kind”. As an example,
he proposes integrating cos(xE) · a0 and sin(xE) · a0 .
O’Kinealy’s improvements are emphatically received by Glaisher in a
short letter [9] to the editors in which he remarks that he had examined
O’Kinealy’s work and that, “after developing the method so far as to include
these formulae and several others, I communicated it, with the examples,
to Professor Cayley, in a letter on the 22nd or 23rd of July, which gave
rise to a short correspondence between us on the matter at the end of July.
(2.2) a0 −a1 x2 +a2 x4 −· · · =
RAMANUJAN’S MASTER THEOREM
3
My only reason for wishing to mention this at once is that otherwise, as I
hope soon to be able to return to the subject and somewhat develop the
principle, which is to a certain extent novel, it might be thought at some
future time that I had availed myself of Mr. O’Kinealy’s idea without proper
acknowledgement.”
Unfortunately, no further work seems to have appeared along these lines
so that one can only speculate as to what Glaisher and Cayley have figured
out. It is not unreasonable to guess that they might very well have developed
an idea somewhat similar to Ramanujan’s Master Theorem (1.1). In fact,
just slightly generalizing O’Kinealy’s argument is enough to formally obtain
(1.1). This is shown next.
Formal proof of (1.1).
Z ∞
Z ∞
∞
∞
X
X
(−1)n n n
(−1)n
n
s−1
λ(n)x dx =
xs−1
E x dx · λ(0)
x
n!
n!
0
0
n=0
n=0
Z ∞
=
xs−1 e−Ex dx · λ(0)
0
Γ(s)
=
· λ(0)
Es
= Γ(s)λ(−s)
where in the penultimate step the integral representation
Z ∞
(2.3)
Γ(s) =
xs−1 e−x dx
0
of the gamma function was employed and the operator E treated as a number. It is this step which renders the proof formal: clearly the coefficient
function λ(n) needs to satisfy certain conditions for the result to be valid.
This will be discussed in Section 3.
The identity
Z ∞
(2.4)
xs−1 ϕ(0) − xϕ(1) + x2 ϕ(2) − · · · dx =
0
π
ϕ(−s),
sin sπ
is given by Ramanujan alongside (1.1) (see [2]). The formulations are equivalent: the relation ϕ(n) = λ(n)/Γ(n + 1) converts (2.4) into (1.1).
The integral theorem (2.1) also appears in the text [8] as Exercise 7 on
Chapter XXVI. It is attributed there to Glaisher.
The exercise asks to show
R∞
(2.1) and to “apply this theorem to find 0 sinxax dx.”
The argument that Ramanujan gives for (1.1) appears in Hardy [14] where
the author demonstrates that, while the argument can be made rigorous in
certain cases, it usually leads to false intermediate formulae which “excludes
practically all of Ramanujan’s examples”.
A rigorous proof of (1.1) and its special case (2.1) was given in Chapter
XI of [14]. This text is based on a series of lectures on Ramanujan’s work
given in the Fall semester of 1936 at Harvard University.
4
RAMANUJAN’S MASTER THEOREM
3. Rigorous treatment of the Master Theorem
The proof of Ramanujan’s Master Theorem provided by Hardy in [14]
employs Cauchy’s residue theorem as well as the well-known Mellin inversion
formula which is recalled next followed by an outline of the proof.
Theorem 3.1 (Mellin inversion formula). Assume that F (s) is analytic in
the strip a < Re s < b and define f by
Z c+∞
1
f (x) =
F (s)x−s ds.
2πi c−i∞
If this integral converges absolutely and uniformly for c ∈ (a, b) then
Z ∞
F (s) =
xs−1 f (x) dx.
0
Theorem 3.2 (Ramanujan’s Master Theorem). Let ϕ(z) be an analytic
(single-valued) function, defined on a half-plane
(3.1)
H(δ) = {z ∈ C : Re z ≥ −δ}
for some 0 < δ < 1. Suppose that, for some A < π, ϕ satisfies the growth
condition
(3.2)
|ϕ(v + iw)| < CeP v+A|w|
for all z = v + iw ∈ H(δ). Then (2.4) holds for all 0 < Re s < δ, that is
Z ∞
π
ϕ(−s).
(3.3)
xs−1 ϕ(0) − xϕ(1) + x2 ϕ(2) − · · · dx =
sin sπ
0
Proof. Let 0 < x < e−P . The growth conditions show that the series
Φ(x) = ϕ(0) − xϕ(1) + x2 ϕ(2) − · · ·
converges. The residue theorem yields
Z c+∞
1
π
(3.4)
Φ(x) =
ϕ(−s)x−s ds
2πi c−i∞ sin πs
for any 0 < c < δ. Observe that π/ sin(πs) has poles at s = −n for n =
0, 1, 2, . . . with residue (−1)n . The integral in (3.4) converges absolutely and
uniformly for c ∈ (a, b) for any 0 < a < b < δ. The claim now follows from
Theorem 3.1.
Remark 3.3. The conversion ϕ(u) = λ(u)/Γ(u + 1) establishes Ramanujan’s Master Theorem in the form (1.1). The condition δ < 1 ensures convergence of the integral in (3.3). Analytic continuation may be employed
to validate (3.3) to a larger strip in which the integral converges. See also
Section 7.
RAMANUJAN’S MASTER THEOREM
5
4. A collection of elementary examples
This section contains a collection of definite integrals that can be evaluated directly from Ramanujan’s Master Theorem 3.2. For the convenience
of the reader, the main theorem in the form (1.1) is reproduced below. Its
hypotheses are described in Section 3.
Theorem 4.1. Assume f admits an expansion of the form
(4.1)
f (x) =
∞
X
λ(k)
k=0
k!
(−x)k .
Then, the Mellin transform of f is given by
Z ∞
xs−1 f (x) dx = Γ(s)λ(−s).
(4.2)
0
Example 4.2. Instances of series expansions involving factorials are particularly well-suited for the application of Ramanujan’s Master Theorem. To
illustrate this fact, use the binomial theorem for a > 0 in the form
∞
∞ X
X
Γ(k + a) (−x)k
k+a−1
−a
(−x)k =
(4.3)
(1 + x) =
.
k
Γ(a)
k!
k=0
k=0
Ramanujan’s Master Theorem (1.1), with λ(k) = Γ(a + k)/Γ(a), then yields
Z ∞ s−1
x
dx
Γ(s)Γ(a − s)
(4.4)
=
= B(s, a − s)
a
(1
+
x)
Γ(a)
0
where B is the beta integral.
Example 4.3. Several of the functions appearing in this paper are special
cases of the hypergeometric function
(4.5)
∞
X
(c1 )k (c2 )k · · · (cp )k (−x)k
p Fq (c; d; −x) =
(d1 )k (d2 )k · · · (dq )k k!
k=0
where c = (c1 , . . . , cp ), d = (d1 , . . . , dq ), and (a)k = a(a + 1) · · · (a + k − 1)
denotes the rising factorial. To apply Ramanujan’s Master Theorem, write
(a)k = Γ(a + k)/Γ(a). The result is the standard evaluation
(4.6)
Z ∞
Γ(c1 − s) · · · Γ(cp − s)Γ(d1 ) · · · Γ(dq )
xs−1 p Fq (c; d; −x) dx = Γ(s)
,
Γ(c1 ) · · · Γ(cp )Γ(d1 − s) · · · Γ(dq − s)
0
which appears as Entry 7.511 in [13].
Example 4.4. The Bessel function Jν (x) admits the hypergeometric representation
1
xν
x2
(4.7)
Jν (x) =
F
−;
ν
+
1;
−
.
0 1
Γ(ν + 1) 2ν
4
6
RAMANUJAN’S MASTER THEOREM
Its Mellin transform is therefore obtained from (4.6) as
Z ∞
2s−1 Γ s+ν
s−1
2 x Jν (x) dx =
(4.8)
.
Γ ν−s
0
2 +1
This formula appears as 6.561.14 in [13].
Example 4.5. The expansion
√
∞
cos(t tan−1 x) X Γ(t + 2k) Γ(k + 1) (−x)k
=
.
Γ(t) Γ(2k + 1)
k!
(1 + x)t/2
k=0
was established in [3] in the process of evaluating of a class of definite integrals (alternatively, as pointed out by the referee, the expansion may be
deduced hypergeometricly; in fact, the conversion is done automatically by
Mathematica 7 upon expressing the series as a hypergeometric function). A
direct application of Ramanujan’s Master Theorem yields
Z ∞
−1 √x)
Γ(2t − 2ν) Γ(1 − ν) Γ(ν)
ν−1 cos(2t tan
dx =
,
x
t
(1 + x)
Γ(2t) Γ(1 − 2ν)
0
and x = tan2 θ gives
Z π/2
sinµ θ cos2t−µ θ cos(2tθ) dθ =
(4.9)
0
πΓ(2t − µ − 1)
.
2 sin(πµ/2) Γ(2t)Γ(−µ)
Similarly, the expansion
√
∞
sin(2t tan−1 x) X Γ(2t + 2k + 1) Γ(k + 1) (−x)k
√
=
Γ(2t)Γ(2k + 2)
k!
x(1 + x)t
k=0
produces
Z
(4.10)
π/2
sinµ−1 θ cos2t−µ θ sin(2tθ) dθ =
0
πΓ(2t − µ)
.
2 sin(πµ/2)Γ(2t)Γ(1 − µ)
Example 4.6. The Mellin transform of the function log(1 + x)/(1 + x) is
obtained from the expansion
∞
X
log(1 + x)
(4.11)
=−
Hk (−x)k ,
1+x
k=1
1+ 21 +· · ·+ k1
where Hk =
is the kth harmonic number. The analytic continuation of the harmonic numbers, required for an application of Ramanujan’s
Master Theorem, is achieved by the relation
(4.12)
Hk = γ + ψ(k + 1),
where ψ(x) = Γ0 (x)/Γ(x) is the digamma function and γ = −Γ0 (1) is the
Euler constant. The expansion (4.11) and Ramanujan’s Master Theorem
now give
Z ∞ ν−1
x
π
(4.13)
log(1 + x) dx = −
(γ + ψ(1 − ν)) .
1
+
x
sin
πν
0
RAMANUJAN’S MASTER THEOREM
The special case ν =
1
2
(4.14)
7
produces the logarithmic integral
Z ∞
log(1 + t2 )
dt = π log 2
1 + t2
0
which is equivalent to the classic evaluation
Z π/2
π
log sin x dx = − log 2
(4.15)
2
0
given by Euler.
Example 4.7. The infinite product representation of the gamma function
∞
x −1 x/n
e−γx Y e
1+
(4.16)
Γ(x) =
x
n
n=1
is equivalent to the expansion
(4.17)
log Γ(1 + x) = −γx +
∞
X
ζ(k)
k=2
k
(−x)k .
Hence Ramanujan’s Master Theorem implies
Z ∞
π ζ(2 − ν)
γx + log Γ(1 + x)
dx =
,
(4.18)
xν−1
2
x
sin
πν 2 − ν
0
valid for 0 < ν < 1.
5. A quartic integral
The authors’ first encounter with Ramanujan’s Master Theorem occured
while evaluating the quartic integral
Z ∞
dx
(5.1)
N0,4 (a; m) =
.
4
(x + 2ax2 + 1)m+1
0
The goal was to provide a proof of the experimental observation that
π
Pm (a),
(5.2)
N0,4 (a; m) = m+3/2
2
(a + 1)m+1/2
where
(5.3)
Pm (a) = 2
−2m
m
X
k=0
2
k
2m − 2k
m−k
m+k
(a + 1)k .
m
The reader will find in [1] a variety of proofs of this identity, but it was
Ramanujan’s Master Theorem that was key to the first proof of (5.2). This
proof is outlined next.
p
√
The initial observation is that the double square root function a + 1 + c
satisfies the unexpected relation
Z ∞
q
√
d
1
dx
(5.4)
a+ 1+c= √
.
4 + 2ax2 + 1 + c
dc
x
π 2 0
8
RAMANUJAN’S MASTER THEOREM
This leads naturally to the Taylor series expansion
q
∞
√
√
1 X (−1)k−1
(5.5)
a+ 1+c= a+1+ √
N0,4 (a; k − 1)ck .
k
π 2 k=1
Thus, in terms of
λ(k) = −
(5.6)
(k − 1)!
√ N0,4 (a; k − 1),
π 2
Ramanujan’s Master Theorem implies that
Z ∞
q
√
s−1
(5.7)
Γ(s)λ(−s) =
c
a + 1 + c dc.
0
The next ingredient emerges from a direct differentiation of the integral N0,4 :
j
Z
d
(−1)j 2j (k + j − 1)! ∞
x4k+2j−2 dx
N0,4 (a; k − 1) =
da
(k − 1)!
(x4 + 2ax2 + 1)k+j
0
Note that the integral on the right-hand side can be expressed in terms of
N0,4 if j = 1 − 2k. In this case, the formal relation
(5.8)
d
da
1−2k
λ(k) = (−2)1−2k λ(1 − k)
is obtained. This may be rewritten as
1 d 2m+1
λ(−m)
(5.9)
λ(m + 1) = −
2 da
and relates the quartic integral N0,4 (a; m), as a function in m, to its analytic
continuation appearing in (5.7). Combining (5.9) and (5.7) one arrives at
√
2m+1 Z ∞
q
√
π 2
d
m−1
N0,4 (a; m) = 2m+1
a + 1 + c dc
c
2
(m − 1)!m! da
0
√ Z ∞
mπ 2 4m 2m
cm−1 dc
√
(5.10)
= 6m+2
.
2
2m
m
(a + 1 + c)2m+1/2
0
√
The substitution u = 1 + c shows that
√ Z ∞
mπ 2 4m 2m
(5.11) N0,4 (a; m) = 6m+1
fm (u)(a + u)−(2m+1/2) du,
2
2m
m
1
with fm (u) = u(u2 − 1)m−1 . This final integral can now be evaluated to give
the desired expression (5.2) for N0,4 . To this end one integrates by parts
and uses the fact that the derivatives of fm at u = 1 have a closed-form
evaluation. Further details can be found in [4].
RAMANUJAN’S MASTER THEOREM
9
6. Random Walk Integrals
In this section, the n-dimensional integral
s
Z
n
X
(6.1)
Wn (s) :=
e2πixk dx1 dx2 · · · dxn
n
[0,1]
k=1
is considered which has recently been studied in [5] and [6]. This integral
is connected to planar random walks. In detail, such a walk is said to be
uniform if it starts at the origin and at each step takes a unit-step in a
random direction. As such, (6.1) expresses the s-th moment of the distance
to the origin after n steps. The study of these walks originated with K.
Pearson more than a century ago [16].
For s an even integer, the moments Wn (s) take integer values. In fact,
for integers k ≥ 0, the explicit formula
2
X
k
(6.2)
Wn (2k) =
a1 , . . . , a n
a1 +···+an =k
has been established in [5]. The evaluation of Wn (s) for values of s 6= 2k
is more challenging. In particular, the definition (6.1) is not well-suited for
high-precision numerical evaluations, and other representations are needed.
In the remainder of this section, it is indicated how Ramanujan’s Master
Theorem may be applied to find a one-dimensional integral representation
for Wn (s). While (6.1) may be used to justify a priori that Ramanujan’s
Master Theorem 3.2 applies, it should be noted that one may proceed formally with only the sequence (6.2) given. This is the approach taken below
in the proof of Theorem 6.1. Ramanujan’s Master Theorem produces a
formal candidate for an analytic extension of the sequence Wn (2k). This
argument yields the following Bessel integral representation of (6.1), previously obtained by D. Broadhurst [7].
Theorem 6.1. Let s ∈ C with 2k > Re s > max(−2, − n2 ). Then
s Z ∞
1 d k n
s+1−k Γ(1 + 2 )
2k−s−1
(6.3)
Wn (s) = 2
x
−
J0 (x) dx.
Γ(k − 2s ) 0
x dx
Proof. The evaluation (6.2) yields the generating function for the even moments:

n
k
k
X
X
√ n
(−x)
(−x) 

(6.4)
Wn (2k)
=
=
J
(2
x) ,
0
(k!)2
(k!)2
k≥0
k≥0
with J0 (z) the Bessel function of the first kind as in (4.7). Applying Ramanujan’s Master Theorem (1.1) to λ(k) = Wn (2k)/k! produces
Z ∞
√
(6.5)
Γ(ν)λ(−ν) =
xν−1 J0n (2 x) dx.
0
10
RAMANUJAN’S MASTER THEOREM
A change of variables and setting s = 2ν gives
Z
Γ(1 − s/2) ∞ s−1 n
(6.6)
Wn (−s) = 21−s
x J0 (x) dx.
Γ(s/2)
0
The claim now follows from the fact that if F (s) is the Mellin transform of
f (x) then (s−2)(s−4) · · · (s−2k)F (s−2k) is the corresponding transform of
d k
− x1 dx
f (x). The latter is a consequence of Ramanujan’s Master Theorem.
7. Extending the domain of validity
The region of validity of the identity given by Ramanujan’s Master Theorem is restricted by the region of convergence of the integral. For example,
the integral representation of the gamma function given in (2.3) holds for
Re s > 0. In this section it is shown that analytic continuations of such
representations are readily available by dropping the first few terms of the
Taylor series of the defining integrand. This provides an alternative to the
method used at the end of the proof of Theorem 6.1.
Theorem 7.1. Suppose ϕ satisfies the conditions of Theorem 3.2 so that
for all 0 < Re s < δ
Z ∞
∞
X
π
s−1
ϕ(−s).
x
ϕ(k)(−x)k dx =
sin sπ
0
k=0
Then, for any positive integer N and −N < Re s < −N + 1,
Z ∞
∞
X
π
(7.1)
xs−1
ϕ(k)(−x)k dx =
ϕ(−s).
sin sπ
0
k=N
Proof. Applying Theorem 3.2 to the function ϕ(· + N ) shows that
Z ∞
∞
X
π
xs−1
ϕ(−s + N ).
ϕ(k + N )(−x)k dx =
sin sπ
0
k=0
Now shift s to obtain (7.1).
Example 7.2. Apply the result (7.1) with N = 1 to obtain
Z ∞
(7.2)
Γ(s) =
xs−1 e−x − 1 dx.
0
This integral representation now gives an analytic continuation of (2.3) to
−1 < Re s < 0.
8. Some classical polynomials
In this section the explicit formulas for the generating functions of classical
polynomials are employed to derive some definite integrals.
RAMANUJAN’S MASTER THEOREM
11
8.1. The Bernoulli polynomials. The generating function for the Bernoulli
polynomials Bm (q) is given by
(8.1)
∞
X
teqt
tm
=
B
(q)
.
m
et − 1
m!
m=0
These polynomials relate to the Hurwitz zeta function
(8.2)
ζ(z, q) =
∞
X
n=0
1
(n + q)z
via Bm (q) = −mζ(1 − m, q) for m ≥ 1. Then (8.1) yields
(8.3)
∞
X
1
(−t)m
e−qt
−
=
ζ(−m,
q)
.
1 − e−t
t
m!
m=0
Ramanujan’s Master Theorem now provides the integral representation
−qt
Z ∞
1
e
ν−1
−
dt = Γ(ν)ζ(ν, q),
(8.4)
t
1 − e−t
t
0
valid in the range 0 < Re ν < 1.
8.2. The Hermite polynomials. The generating function for the Hermite
polynomials Hm (x) is
(8.5)
2
e2xt−t =
∞
X
m=0
Hm (x)
tm
.
m!
Their analytic continuation, as a function in the index m, is given by
m 1 2
m
(8.6)
Hm (x) = 2 U − , , x
2 2
where U is Whittaker’s confluent hypergeometric function. Ramanujan’s
Master Theorem now provides the integral evaluation
Z ∞
Γ(s)
s 1 2
s−1 −2xt−t2
, ,x .
(8.7)
t e
dt = s U
2
2 2
0
An equivalent form of this evaluation appears as Entry 3.462.1 in [13].
8.3. The Laguerre polynomials. The Laguerre polynomials Ln (x) given
by
X
∞
1
xt
(8.8)
exp −
=
Ln (x)tn
1−t
1−t
n=0
can be expressed also as Ln (x) = M (−n, 1; x), where
X
∞
(a)j xj
a (8.9)
M (a, c; x) = 1 F1
x
=
c
(c)j j!
j=0
12
RAMANUJAN’S MASTER THEOREM
is the confluent hypergeometric or Kummer function. Ramanujan’s Master
Theorem yields the evaluation
Z ∞ ν−1
t
xt
(8.10)
dt = Γ(ν)Γ(1 − ν)M (ν, 1; x).
exp
1+t
1+t
0
The change of variables r = t/(1 + t) then gives
Z 1
1
(8.11)
M (ν, 1; x) =
rν−1 (1 − r)−ν erx dr,
Γ(ν)Γ(1 − ν) 0
which is Entry 9.211.2 in [13].
(α,β)
8.4. The Jacobi polynomials. The Jacobi polynomials Pn
fined by the generating function
(8.12)
∞
X
Pn(α,β) (x)tn =
n=0
(x) are de-
2α+β
(1 − t + R∗ (x, t))−α (1 + t + R∗ (x, t))−β ,
R∗ (x, t)
√
where R∗ (x, t) = 1 − 2xt + t2 . These polynomials admit the hypergeometric representation
Γ(n + 1 + α)
1−x
(8.13) Pn(α,β) (x) =
F
n
+
α
+
β
+
1,
−n;
1
+
α;
.
2 1
n! Γ(1 + α)
2
√
Now write R(x, t) = R∗ (x, −t), so that R(x, t) = 1 + 2xt + t2 , to obtain
(8.14)
2α+β R−1 (1 + t + R)−α (1 − t + R)−β =
∞
X
λ(k)
k=0
(−t)k
k!
where
(8.15)
Γ(k + 1 + α)
1−x
λ(k) =
.
2 F1 k + α + β + 1, −k; 1 + α;
Γ(1 + α)
2
Ramanujan’s Master Theorem produces
Z ∞
tν−1 dt
α
β
0 R(1 + t + R) (1 − t + R)
B(ν, 1 + α − ν)
1 + α + β − ν, ν 1 − x
=
.
2 F1
2
1+α
2α+β
8.5. The Chebyshev polynomials of the second kind. These polynomials are defined by
sin((n + 1)x)
,
sin x
and have the generating function
(8.16)
(8.17)
Un (a) =
∞
X
k=0
Uk (a)xk =
where cos x = a,
1
.
1 − 2ax + x2
RAMANUJAN’S MASTER THEOREM
13
The usual application of Ramanujan’s Master Theorem yields
Z ∞
xν−1 dx
π sin[(1 − ν) cos−1 a]
√
(8.18)
=
.
1 + 2ax + x2
sin πν
1 − a2
0
This result appears as Entry 3.252.12 in [13].
9. The method of brackets
The focus of this final section will be on a multi-dimensional extension of
Ramanujan’s Master Theorem. This has been called the method of brackets
and it was originally presented in [12] in the context of integrals arising from
Feynman diagrams. A complete description of the operational rules of the
method, together with a variety of examples, was first discussed in [11]. The
basic idea is the assignment of a formal symbol hai to the divergent integral
Z ∞
(9.1)
xa−1 dx.
0
The rules for operating with brackets are described below. These rules
employ the symbol
(9.2)
φn =
(−1)n
,
Γ(n + 1)
called the indicator of n.
Rule 1. The bracket expansion
X
1
mr hα + m1 + · · · + mr i
1
=
φm1 ,...,mr am
1 · · · ar
α
(a1 + a2 + · · · + ar )
Γ(α)
m ,...,m
1
r
holds. Here φm1 ,...,mr is a shorthand notation for the product φm1 · · · φmr .
Where there is no possibility
of confusion this will be further abridged as
P
φ{m} . The notation {m} is to be understood likewise.
Rule 2. A series of brackets
X
φ{n} f (n1 , . . . , nr ) ha11 n1 + · · · a1r nr + c1 i · · · har1 n1 + · · · arr nr + cr i
{n}
is assigned the value
1
f (n∗1 , · · · , n∗r )Γ(−n∗1 ) · · · Γ(−n∗r ),
|det(A)|
where A is the matrix of coefficients (aij ) and (n∗i ) is the solution of the
linear system obtained by the vanishing of the brackets. No value is assigned
if the matrix A is singular.
Rule 3. In the case where a higher dimensional series has more summation
indices than brackets, the appropriate number of free variables is chosen
among the indices. For each such choice, Rule 2 yields a series. Those
converging in a common region are added to evaluate the desired integral.
14
RAMANUJAN’S MASTER THEOREM
Example 9.1. Apply the method of brackets to
Z ∞
xν−1 F (x) dx
(9.3)
0
where F has the series representation
F (x) =
∞
X
φk λ(k)xk .
k=0
Then (9.3) can be written as the bracket series
Z ∞
Z ∞X
∞
X
xν−1 F (x) dx =
φk λ(k)xk+ν−1 dx =
φk λ(k) hk + νi .
0
0
k=0
k
Rule 2 assigns the value
X
(9.4)
φk λ(k) hk + νi = λ(k ∗ )Γ(−k ∗ )
k
where
k∗
is the solution of k + ν = 0. Thus one obtains
Z ∞
xν−1 F (x) dx = λ(−ν)Γ(ν).
(9.5)
0
This is precisely Ramanujan’s Master Theorem as given by Theorem 3.2.
Rule 1 is a restatement of the fact that the Mellin transform of e−x is
Γ(s):
Z ∞
Γ(s)
=
xs−1 e−(a1 +...+ar )x dx
(a1 + . . . + ar )s
0
Z ∞
r X
Y
φmi (ai x)mi dx
=
xs−1
0
=
X
i=1 mi
mr
1
φ{m} am
1 · · · ar hs + m1 + · · · + mr i .
{m}
Example 9.1 has shown that the 1-dimensional version of Rule 2 is Ramanujan’s Master Theorem. A formal argument is now presented to show
that the multi-dimensional version of Rule 2 follows upon iterating the onedimensional result. The exposition is restricted to the 2-dimensional case.
Consider the bracket series
X
(9.6)
φn1 φn2 f (n1 , n2 ) ha11 n1 + a12 n2 + c1 i ha21 n1 + a22 n2 + c2 i
n1 ,n2
which encodes the integral
Z ∞Z ∞ X
φn1 φn2 f (n1 , n2 )xa11 n1 +a12 n2 +c1 −1 y a21 n1 +a22 n2 +c2 −1 dx dy.
0
0
n1 ,n2
RAMANUJAN’S MASTER THEOREM
15
dudv
1
Substituting (u, v) = (xa11 y a21 , xa12 y a22 ) yields dxdy
xy = |a11 a22 −a12 a21 | uv ,
and hence the above integral simplifies to
Z ∞Z ∞ X
1
∗
∗
φn1 φn2 f (n1 , n2 )un1 −n1 −1 v n2 −n2 −1 du dv.
|a11 a22 − a12 a21 | 0
0 n ,n
1
2
(n∗1 , n∗2 )
Here
is the solution to a11 n∗1 +a12 n∗2 +c1 = 0, a21 n∗1 +a22 n∗2 +c2 = 0.
Ramanujan’s Master Theorem gives
Z ∞X
∗
φn1 f (n1 , n2 )un1 −n1 −1 du = f (n∗1 , n2 )Γ(−n∗1 ).
0
n1
A second application of Ramanujan’s Master Theorem shows that the bracket
series (9.6) evaluates to
1
f (n∗1 , n∗2 )Γ(−n∗1 )Γ(−n∗2 ).
|a11 a22 − a12 a21 |
This is Rule 2.
9.1. A gamma-like higher dimensional integral. The next example
illustrates the power and ease of the method of brackets for the treatment
of certain multidimensional integrals such as
Z ∞
Z ∞
n
Y
α
···
exp (−(x1 + . . . + xn ) )
(9.7)
xisi −1 dxi .
0
0
i=1
It should be pointed out that this class of integrals is beyond the scope of
current computer algebra systems including Mathematica 7 and Maple 12.
For simplicity of exposition, take n = 2 in (9.7). The n-dimensional case
presents no additional difficulties.
Z ∞Z ∞
xs−1 y t−1 exp (−(x + y)α ) dx dy
0
0
X Z ∞Z ∞
=
φj
xs−1 y t−1 (x + y)αj dx dy
0
j
=
X
Z
φj
j
=
0
∞Z
X
j,n,m
0
φj,n,m
0
∞
xs−1 y t−1
X
n,m
φn,m xn y m
hn + m − αji
dx dy
Γ(−αj)
1
hn + m − αji hn + si hm + ti
Γ(−αj)
Solving the linear equations for the vanishing of the brackets gives n∗ = −s,
m∗ = −t, and j ∗ = − s+t
α . The determinant of the system is α, therefore the
integral is
1
1
1 Γ(s)Γ(t)
s+t
∗
∗
∗
Γ(−n )Γ(−m )Γ(−j ) =
Γ
.
α Γ(−αj ∗ )
α Γ(s + t)
α
The full statement of this result is presented as the next theorem.
16
RAMANUJAN’S MASTER THEOREM
Theorem 9.2.
Z
∞
Z
···
0
∞
exp (−(x1 + . . . + xn )α )
0
n
Y
xsi i −1 dxi
i=1
1 Γ(s1 )Γ(s2 ) . . . Γ(sn )
=
Γ
α Γ(s1 + . . . + sn )
s1 + . . . + sn
α
.
Remark 9.3. The correct interpretation of Rule 3 is work in-progress. The
next example illustrates the subtleties associated with this question. The
evaluation
Z ∞
Γ(s)
xs−1 e−2x dx = s
(9.8)
2
0
follows directly from the bracket expansion
Z ∞
X
xs−1 e−2x dx =
φn 2n hn + si
0
n
and Rule 2. On the other hand, rewriting the integrand as e−2x = e−x e−x
and expanding it in a bracket series produces
Z ∞
X
xs−1 e−x e−x dx =
φn,m hn + m + si .
0
n,m
The resulting bracket series has more summation indices than brackets. The
choice of n as a free variable, gives m∗ = −n − s and Rule 2 produces the
convergent series
∞
X
s (−1)n
Γ(s)
Γ(n + s) = Γ(s)1 F0
(9.9)
−1 = s .
n!
−
2
n=0
Symmetry dictates that the choice of m as a free variable leads to the same
result. Rule 3, as stated currently, would yield the correct evaluation (9.8),
twice.
The trouble has its origin in that the series in (9.9) has been evaluated
at the boundary of its region of convergence. Rule 3 should be modified by
introducing extra parameters to distinguish different regions of convergence.
This remains to be clarified. For instance,
Z ∞
X
(9.10)
xs−1 e−Ax e−Bx dx =
φn,m An B m hn + m + si
0
n,m
which, upon choosing n and m as free variables, yields the two series
Γ(s)
s A
Γ(s)
s B
−
−
,
1 F0
1 F0
Bs
− B
As
− A
respectively. Both series evaluate to Γ(s)/(A + B)s , but it is now apparent
that their regions of convergence are different. Accordingly, they should not
be added in order to obtain the value (9.10). The original integral (9.8)
appears as the limit A, B → 1.
RAMANUJAN’S MASTER THEOREM
17
Acknowledgements. The fifth author wishes to thank the partial support
of NSF-DMS 0070757. The work of the last author was partially supported,
as a graduate student, by the same grant.
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18
RAMANUJAN’S MASTER THEOREM
Department of Mathematics, Tulane University, New Orleans, LA 70118
E-mail address: [email protected]
Departmento de Fisica, Universidad Santa Maria
E-mail address: [email protected]
Prestadigital LLC, 12114 Kimberly Lane, Houston, TX
E-mail address: [email protected]
Department of Mathematics, Tulane University, New Orleans, LA 70118
E-mail address: [email protected]
Department of Mathematics, Tulane University, New Orleans, LA 70118
E-mail address: [email protected]