EXPANDING THE GAMMA FUNCTION NEAR ITS NEGATIVE

!
!
EXPANDING THE GAMMA FUNCTION NEAR ITS NEGATIVE
SINGULARITIES
Andrei Vieru
!
Abstract
We propose a general method to find the coefficients of the Laurent series of the
Gamma function near any of its negative poles, avoiding any occurrence of Digamma
or Polygamma functions. Unfolding the algorithm to arbitrarily chosen –n and to
arbitrarily high-rank terms, a nice self-similarity arises in the analytical expression of
these coefficients. The appearance of 1/n! in the Laurent expansion near –n is but one
more argument in favor of Legendre normalization of the Gamma function.
!
Keywords: Euler constant, harmonic numbers, Gamma function, Legendre
normalization,
!
1.1. Basic notations
Euler’s constant is defined as
(1)
Its relation to the Gamma function is well-known (via the Digamma function):
(2)
where
!
Euler’s constant has a direct connection with the gamma function as it can be seen in
the limit formula:
(3)
!
Euler’s constant is related to the set of Harmonic numbers by its very
definition and is related to each Harmonic number in particular by the formula:
(4)
!1
It is also related to the set of fractional Harmonic numbers by the formula:
!
!
!
!
!
Euler’s limit definition of the gamma function
works directly in the whole complex domain without need of analytic continuation.
!
2. In search of Harmonic numbers near Gamma singularities
2.0. The neighborhoods of zero
Changing in (3) the bounded variable x into 1/x gives
(5)
where x may approach 0 from any side.
2.1. The neighborhoods of –1
Assume x ≠ 0. Then
(6)
!
Here the right-handed side contains the first Harmonic number – H
!
1
= –1
2.2. The neighborhoods of other negative integers
(7)
The statement in its general form is:
!
!2
!
!
For any non negative integer n
!
!
!
!
and therefore
(8)
!
!
!
!
which (after elementary algebraic manipulation) might be considered as a
generalization of (5) limit formula, provided we consider H0 = 0 by virtue of Euler’s
integral representation
!
3. Strictly negative singularities
!
Using strictly negative points of singularity of the gamma function, Euler’s
constant may also be expressed in the following way: for any integer greater than 0
!
or
!
!3
For n = 0 the following limit formula holds:
!
!
!
!
!
Surprisingly,
(In other words, the rhs in (♠) may be absorbed in its lhs, and then, dividing
by (– 1)nx, one gets the same limit. Note that n!+(– 1)n+1 has to be embedded twice in
the rhs of the new limit formula.)
!
One can unify (♠) and (♣) in one single formula:
!
'
*
1
ε n0
−
Γ
−n
+
x
0
(
)
)
,
1
2
n +1
x
2 (γ − H n )
χn0 +
= (−1) //
)
,
δ
n +1 2
! lim
n0
x →0
x
−1− (−1) xΓ(−n − x ) ,
. n!+ χ n 0 (−1) 1
)
(
+
!
{0 if
! where χ i j = 1 if
€
€
€
i= j
−1 if i= j
, δi j is the Kronecker delta symbol and εi j =
i≠ j
1 if i≠ j
{
! (in the context χ n 0 = 0 if n=0 is equivalent to the Dirac measure of the set N+ )
1 if n≠0
!
{
The presence of Kronecker-like symbols in the unified formula points out to a
specificity of the rightmost point of singularity of the gamma function (zero).
!
!
4. EXPANDING THE GAMMA FUNCTION NEAR ITS NEGATIVE POLES
!
4.1. Trying to expand gamma function near its singularities
!
In (4°), page 297, the authors comment Ramanujan’s attempt to expand gamma
function for positive values greater than 1 and show to which extent Ramanujan’s
!4
computations are accurate. Curiously, none of them present the second coefficient in a
closed form. 0.9890559953… is the evaluation of
!
!
!
and has closed form
The expansion of the Gamma function near 0 is well-known:
!
!
!
!
We shall not try to do better for positive values and not even for negative values near
0. As we have already seen, this singularity is somewhat different from all other
strictly negative poles of Gamma function. This can be seen comparing the following
formula:
!
!
!
(10)
with its analogues for the negative poles. The Iohannis formula states that for any
positive integer n
!
!
Therefore, the first three terms of the Laurent series near any pole –n are (for z with
small modulus):
(11°)
(in the rhs of the Iohannis formula, the sum
!
!
!5
shows up ‘from
thin air’ and it is difficult to consider it equals zero when n equals zero)
It is somewhat trickier to find the next term of the Laurent series expansion in a
closed form that will fit for all negative poles.
Here it is:
(12°)
!
The nice thing is that the coefficients of the Laurent series expansion around strictly
negative poles display some kind of ‘grammatical fractality’:
!
!
in the third coefficient appears as a part of
!!
!
in the fourth coefficient
!
!
!
The self-similarity is obvious. It will become explicit at the end of this text.
!!
4.2. Two classical recursive formulas
!
The coefficients of the Taylor expansion of the reciprocal Gamma function obey to
the following recursive formula, valid for n > 2:
with
!
!
and
How and why the Riemann zeta function at positive integer values appears in this
context is well explained in (6°).
To get a similar recursive formula for the coefficients of the Laurent expansion of the
Gamma function itself (near 0), one has to change signs:
!
here we have
!
and
Τhe recursive formula which provides the coefficients of the Laurent expansion of the
Gamma function near 0 becomes:
!6
!
!
!
!
The numerical values of the first four coefficients appear in Ramanujan’s second
notebook.
Their accuracy is commented in (4°). The authors propose more accurate values than
Ramanujan’s ones, but they manage to print still one or two inaccurate digits in both
b4 and b5 (i.e., following their notation, b3 and b4). Neither of them propose the
following closed forms (I switch to my indexes):
!
!!
!
!
!
!!
!
As we have already seen, when we expand Gamma function near its negative
singularities, we get, along with the factorials, coefficients whose expressions contain
also Harmonic numbers. They first show up in simple expressions of the form:
!!
then
later on:
i.e.
– b2 – Hn
!!
!! i.e.
!
!!
!!
!!
!
!7
the next step:
!
!
!
!
!!
!
!
!
!
And so on…
On the basis of these ‘nested sums’, one can imagine the complete solution of
the problem of expanding Gamma function near each of its negative poles.
!
Choose a negative pole – n, compute as many Harmonic numbers as wanted,
compute as many bi as needed, restore the missing factorials (near the pole –n, every
term of the Laurent series has a factor (–1)n+m/n!, as can be seen in (11°) and (12°)),
and one arrives at an arbitrarily long and precise beginning of the asymptotic series.
!!
Andrei Vieru
e-mail: [email protected]
!!
!!
REFERENCES
!(1°) Jonathan Sondow “CRITERIA FOR IRRATIONALITY OF EULER'S CONSTANT”,
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY, Volume 131, Number
11, Pages 3335-3344
!(2°) Julian Havil “GAMMA EXPLORING EULER'S CONSTANT”, PRINCETON
UNIVERSITY PRESS, 2003
!(3°) Xavier Gourdon and Pascal Sebah, COLLECTION OF FORMULAE FOR EULER’S
CONSTANT
!(4°) BRUCE C. BERNDT, ROBERT L. LAMPHERE, AND B. M. WILSON CHAPTER 12
OF RAMANUJAN'S SECOND NOTEBOOK: CONTINUED FRACTIONS
!(5°) L. BOURGUET, SUR LES INTEGRALES EULERIENNES ET QUELQUES AUTRES
FONCTIONS UNIFORMES, ACTA MATHEMATICA, VOL. 2, PP. 261-295, (1883).
!
(6°) H. Kleinert and V. Schulte-Frohlinde, REGULARIZATION OF
FEYNMAN INTEGRALS http://users.physik.fu-berlin.de/~kleinert/b8/psfiles/08.pdf (in
‘CRITICAL PROPERTIES OF Φ4-THEORIES’)
!8