A GENERALIZED FORMULA OF HARDY

We give new formulae applicable to the theory of partitions. Recent work suggests they also relate to quasi-crystal structure and self-similarity. Other recent work has given continued fractions for the type of functions herein. Hardy originally gave such formulae as ours in early work on gap power series which led to his and Littlewood's "High Indices" Theorem. Over a decade ago, Mahler and then others proved results on irrationality of decimal fractions applicable to types of functions we consider.

In recent years interest has arisen in the continued fraction forms of series such as f(z) of (1.4).For an account of such work see [4,23,30,32,33,36,37].

SOME NEW RESULTS.
As an undergraduate student in 1979 the author gave the following formula (see [7]), which has Hardy's formula (1.2) as the case m 0. THEOREM 2.1.For non-negative integers m, Re z 0, Re a > 1, --aka )71 (2.1.1) 'L (m+l where F () is the 3-th derivative of the gamma finction, B,, the m-th Bernoulli number, X,' is the sum over nonzero integers k, and k .... at k rn O.
(iv) It can be seen from the above, that (2.1.4)is a sum of m + oscillating functions, each multiplied by a power of log x.Each oscillating function is of order o(1) and oscillates between finite limits.In Hardy's original paper [9] estimates were given for these limits in terms of a.In Mahler [201 an elementary method for obtaining such estimates was also given.However, if a 2 in (1.2) it is easy to use a small calculator to establish that the oscillatory term only enters in after about the fourth decimal place, so for a 2, o(0.001) seems a conservative estimate for the oscillatory function under 3.
PROOFS OF THEOREM 2.1.
The first proof of Theorem 2.1 appeared in Campbell [7].It depended on Lemma 3.1 below, and comparing partial derivatives with respect to the different variables, then comparison to obtain a constant of integration (2.1.3).This was somewhat of a departure from the approach taken by Hardy, since the method in [7] required distinct to a as an integer greater than unity.(1.1) as it appears in Hardy's book on Ramanujan [12] actually has a misprint indicating the author was not thinking of a continuous variable a.
The landmark paper of Hardy and Littlewood [13] contains theorems of sufficient generality to justify all of the differentiations of the series of kind (2.1.1)as they occur in [7] and in the present note.(1.1) is a case of the following.Lemma 8.1.I/'Rez > 0, Rea > 1, Ren > 0, }=-ak'e z" loga r(n)- (3.1) PROOF.This comes easily from both straightforward application of the r ue theorem, and by summing on the Mellin inversion formula, so the integral 2ri r(-=)*" (x a==)-I d= PROOF OF THEOREM 2.1.We consider the behaviour of the terms in the bilateral series on the left side of (3.1) as n 0. In the direction with positive k terms the series clearly converges, however for negative k terms the series from both sides of (3.1) so that (3.4) becomes the convergent E (e--1) k--1 when n 0, and we are left with Hardy's formula (1.2), after calculating the corresponding limit for the right side of the compensated version of (3.1).Next, we see that if (3.1) be written in the form + a-(e "-1) k--0 k=l __r() x kr + ig-g/ (3.) z" log a 1 a " log a we can formy expand each side into power series in n and equate coefficients to ve at the theorem.This is justified since (3.5) holds for any n > 0 in the neighbourhood of n 0.
SKETCH OF THE EARLIER UNPUBLISHED PROOF.Whilst the proof in Camp- bell [7] is longer and more laboured, it has the merit of showing an interplay between the inde- pendent variables z, a, and n.It starts with our Lemma 3. Integration with respect to z gives the major terms of Theorem 2.1, with the constant term (2.1.3)obtained from the functional equation /" (' =/"1 E "= f (' "1 clearly satisfied by the series (2.1.1).Of course the crucial part of all this is Lemma 3.1, and neither of the two proofs could stand without it.
Theorem 2.1 and indeed Lemma 3.1 itself may be used to derive identities for generating functions of partitions of various sorts into a-th powers.To begin just with (1.2), we can easily show that THEOREM 4.1.H Rea > 1, and Re > 0 with z e -t, We note that as a corollary, if a 2 we have the elementary result IIOOF OF THEOREM 4.1. (4.1)may be known, since it is so easily accessible by (1.2).
The proof involves application of the operation Z (--I)a+I f(a, zj)j-1 3--1 to (1.2), letting either side of (1.2) be f(a, ).This, for the left side gives the logarithm of the left side of (4.1).For the right side the same procedure leads to the required result if we know that Z (-1)+a log_k ! (log2) -'r log2 k=l for real n :/: 0. (4.3) is given in Hardy's note on Vacca's series [10].A little simplification thereafter gives the theorem.
Just as Theorem 2.1 generalizes Hardy's formula (1.2), the following formula, which appears to be new, generalizes Theorem 4.1.
(ii) Estimates as z 0 for the function H 2 (4.9) k=l are easily obtained, together with appropriate inequalities, enabling us, if we choose, to focus on /4./.
(iii) Some modification of Mahler's method [20] may be applicable to Theorem 4.2, thus giving elementary estimates for the functions concerned.
(vi) As a curiosity, if z is, say, equal to log 10, the decimal fraction obtained from (4.8) is trivially calculated, and the corresponding (.9) type function can be found easily using a small calculator.The oscillating functions are not so easily calculated, but the major terms of are, as are the order of each of the rn + 1 oscillating functions in G2(z).