Abstract
We used the MACH2 supercomputer to study coefficients in the q-series expansion of , for all . As a result, we were able to conjecture some periodic properties associated with the before unknown location of the maximum coefficient of these polynomials with odd n. Remarkably, the observed period is 62,624.
1. Introduction
Let the q-Pochhammer symbol be (1.1) (1.1) for any Euler’s famous pentagonal number theorem [Citation1] states that infinite product can be written as a sparse power series with all of its coefficients from the set More precisely:
Theorem 1.1.
(Euler’s Pentagonal Number Theorem, 1750). (1.2) (1.2)
In contrast, it is easy to check that the coefficients of the finite products for do not share this property and attain values larger than 1 in size. The authors [Citation2] recently proved that the only 's where the series coefficients stay in the set are when and 5. The first maximums of the absolute value of the coefficients are (1.3) (1.3) From now on we will use maximum absolute coefficient in place of the wordy “the maximum of the absolute value of the coefficients” to simplify our language.
In general, when the coefficients of are studied, we see tame coefficients at both ends of the polynomial and a bubble of oscillation involving enormous integers in the middle. We plot the ordered pairs in , to show the shape of This is to give an idea of the nature of the coefficients of Note that “tame” coefficients are small only relative to the coefficients that appear at the oscillation. For example, in , the 5000th coefficient of seems to be close to the x-axis, but its value is –7,983,490.
Moreover, Sudler [Citation3,Citation4] showed that the maximum absolute coefficients’ size grows exponentially with n: (1.4) (1.4) where,
Sudler and Mr. A. Hurwitz from University of California, Los Angeles calculated On top of that, Wright [Citation5] proved that
Although research on the size of these coefficients existed [Citation3,Citation4, Citation6], there has not been an extensive study of the location of the maximum absolute coefficients of in the literature. When one checks the sequence Equation(1.3)(1.3) (1.3) in the Online Encyclopedia of Integer Sequences they are greeted with the open problem [Citation7] (paraphrased):
If n is even, then is the absolute value of the coefficients of and If n is odd, it is an open question as for which i, where and are the classical floor and ceiling functions.
In this work we present an experimental answer to the question regarding the location of the maximum absolute coefficients of for odd n. Let L(n) denote the lowest exponent i of q in the polynomial, where the coefficient of qi is the maximum absolute coefficient Then, for example, we claim the following.
Conjecture 1.2.
For any , we have
The locations of the largest absolute value of the coefficients for all n as well as the maximum absolute coefficients for all are carefully calculated and verified. This task required painstaking attention to detail, a lot of patience, and a grand computing power for today’s standards. The supercomputer MACH2 was used for this task, which is the largest European installation of its kind. Even with this computing power, the computation was challenging due to the number and the size of the objects that needed to be stored and manipulated at full precision. For perspective, the degree of the polynomial is 2,812,537,500 and the absolute maximum coefficient has 6,465 digits.
In Section 2, we briefly mention how the calculations are carried out. All the main results and our conjectures are presented in Section 3. In Section 4 we shortly discuss some possible future research directions and some related conjectures that appeared in the literature.
2. A brief note on the calculations
To save memory in calculations, one needs to observe that as defined in Equation(1.1)(1.1) (1.1) , satisfies the relation (2.1) (2.1) Using Equation(2.1)(2.1) (2.1) , one can reduce the storage to only half of the coefficients of Another point is that the multiplication to get from is the same as the addition (2.2) (2.2) for all on the coefficient level. Finally, we note that the first n – 1 coefficients of do not change when multiplied with Moreover, they coincide with the first n – 1 coefficients of Equation(1.2)(1.2) (1.2) . Since the maximum absolute coefficients grow Equation(1.4)(1.4) (1.4) , they would not appear in the first n – 1 coefficients that are ±1 or 0.
The task of openly multiplying and locating the maximum of the absolute values of the coefficients started on a reasonably sized RISC server using Maple. One can compute up to a commendable n = 10000 by multiplying these objects using a modern desktop computer. In our experience, the Maple gives up around this point. With Mathematica, we were able to take our calculations to around 15000. This is more or less the upper limit of what computer algebra systems can handle on a today’s consumer level computer. Once this natural boundary was reached we moved on to programing our own versions of the coefficient level multiplication algorithm; first this was done with Python, later with C++. Python does arbitrary precision arithmetics automatically, and Gnu MultiPrecision Library [Citation8] was used for handling large integers in C++. The calculations were carried on the MACH2 supercomputerFootnote1.
3. The location of the maximum absolute coefficients
Let L(n) denote the lowest exponent i of q in the polynomial, where the coefficient of qi is the maximum absolute coefficient When explicitly expanding the polynomials the maximum of the absolute value of the coefficients starts to appear in one or two places consistently for We would like to start by acknowledging that, for all or, in other words, the maximum absolute coefficients of appear as the coefficient(s) of and Now with our explicit computations, this claim (which appears in the Online Encyclopedia of Integer Sequences A160089) extends to all even
For 0 modulo 4 cases the maximum absolute coefficients are the absolute maximum, and for 2 modulo 4 cases the maximum absolute coefficients are the negative of the absolute minimum of For the odd n, both the maximum of the absolute value of the coefficients and its negative appear in due to Equation(2.1)(2.1) (2.1) . Furthermore, we make the claim:
Conjecture 3.1.
For , if , absolute maximum occurs only once as the coefficient of some qi, where and if the absolute maximum occurs only once as the coefficient of some qi, where
There are some initial cases where the maximum absolute coefficient appears as the polynomial coefficients of more than twice. For example, in the n = 33 case the absolute maximum coefficient 56 appears as the coefficients of q270 and q272 and the absolute minimum coefficient –56 appears as the coefficients of q289 and q291, so the maximum absolute coefficient appears a total of 4 times as a coefficient of the polynomial We see that for the maximum absolute coefficient appears as coefficients of only two terms in the q-series expansion of the rising factorial.
We will split the cases for the odd values of n in two groups modulo 4. Before doing so, we define the Dn as the half difference between the two locations of the maximum absolute coefficients for a given odd This is the same as saying how far off the maximum absolute coefficient is from the half degree of the polynomial Hence, knowing this value and n is enough to recover the location of the maximum absolute coefficients. Also we define the canonical difference for all odd
Let be an odd integer, and assume that all necessary Ei and a Dm value for some with modulo 4 are given. Then, one can recover the location of the maximum absolute coefficients for as (3.1) (3.1) where is defined as in Equation(1.1)(1.1) (1.1) .
The first observation we have made and confirmed for is:
Conjecture 3.2.
For all odd . Furthermore, for all odd
By looking at the values and for 1 and 3 modulo 4, we list En’s. The patterns starting from these beginning points look highly periodic with period 19, although this is sadly not the case. The 19 length patterns of 1’s and 2’s that En’s change slightly as the calculations are carried. We match the 19 length patterns that the calculations yield with letters and form an alphabet with 20 letters. The letters that we see in the calculations and their 19 consecutive En value equivalents are given in .
Note that each letter represents 19 consecutive En values (1 or 2) written together without commas to keep the notation simple. The calculation of any letter requires the expansion of 19 × 4 = 76 consecutive polynomials and finding the location of the maximum absolute coefficient and its location of each odd index n. This alphabet makes it possible to find the individual En values. Furthermore, together with Equation(3.1)(3.1) (3.1) , this makes it possible to find the locations of the maximum absolute coefficients. Also note that a and k are symmetric with respect to their centers and a is also special that it is the only letter with six 2 values, whereas all the other words come with five 2 values. It is easy to see that the letters and j read from right to left are the letters and l, respectively.
These letters are observed to be generating almost periodic words starting from and for 1 and 3 modulo 4 values, respectively. We will write exponents of the letters defined in to indicate the number of times a letter has occurred consecutively. From the said starting point we see that the 1 modulo 4 differences give rise to the string
We see that the letter a appears a single time in each word. This phenomenon is the same for the 3 modulo 4 case. We break this string into words for 1 and 3 mod 4 to start every line with an a and write the words in ; 1 and 3 mod 4 cases on the left side and the right side, respectively. In , we also include a line to break period blocks of the size 11 × 20 box both odd residue classes mod 4.
What one can observe from the encoding of is that there is the base word (3.2) (3.2) in each line and an overlying, moving perturbation. It is observed that the first line of the 1 modulo 4 case does not fit any period block and stays as an outlier. We interpret this as the asymptotic behavior of the perturbation kicking in action a little later than the 3 modulo 4 case. More importantly, this perturbation is observably periodic. To express the observed structure better, we represent the base word (3.2) as a vector, where we write the frequencies of the letters as a vector of 20 entries (3.3) (3.3) Then can be expressed as adding some perturbation vectors to B. We define the perturbation vectors for the 1 and 3 modulo 4 classes side-by-side in .
Notice that and are the only two vectors that have two 1’s and all the rest include three 1’s. The first component is always 0. There is a clear pattern in the perturbations and how the motion of the perturbation is from one vector to the following. We have indicated the observed pattern of 1 or 2 repetitions of the perturbations using red and blue (resp.) in . Both in the 1 and 3 mod 4 cases, there are 6 red 1’s which correspond to a letter’s extra appearance as perturbance only a single time in the period. The blue 1’s in the components indicate that the corresponding letter occurs as a part of the perturbation twice. After these observations, the periodicity of this perturbation becomes clearer in vector form.
We claim the following:
Conjecture 3.3.
For any , the i-th and j-th row that appears in and beyond can be associated with the vectors and for 1 and 3 modulo 4 words, respectively.
Moreover, we claim, with these two essential periods for the perturbation, that we can locate the maximum absolute coefficients’ locations for any with odd n.
The base word associated with B has 72 letters, each letter representing 19 consecutive En values, where each of these values are an additional 4 index n shifts of There are 11 base words corresponding to both odd residue classes mod 4. This translates to shifts of the n of by On top of that, the total amount of perturbation is 32 letters, reflecting another 2,432 to the shifts of the subindex of respectively. All together, the period of En’s is conjectured to be 62,624. Lastly, the sum of En’s is 19,787. Now we can start stating our conjectures for the maximum absolute coefficients’ location, using Equation(3.1)(3.1) (3.1) . Our first example is Conjecture 1.2 that is represented in a simplified fachion in the introduction.
Conjecture 1.2.
We have, for any
Note that and Although these formulas might seem limited, one can easily use these explicit values with En’s as represented in to find the location of the maximum coefficient of any odd value greater than 5909. The location calculations for the already exist in the Online Encyclopedia of Integer Sequences, under the sequence number A160089 [Citation7], and one can easily recover the gap between 5000 and 5909, by our above mentioned calculations.
We list similar conjectures that correspond to the start of each word of the periods of 1 and 3 modulo 4 cases:
Conjecture 3.4.
For any , let where is the Kronecker delta and yields 1 if , and 0, otherwise. Then, where the full list of needed seed values of Dn’s are
In Conjecture 3.4, A(k, r) and B(k, r) are used to move to the start of any word. The variable r is used to move within the periods, and k is used for going over full periods. Conjecture 3.4 reduces to Conjecture 1.2 for r = 0. One important note we need to make is the use of the correctional δ functions. The number 76, as explained after the introduction of the chosen alphabet, corresponds to the number of consecutive n’s needed to expand for forming a single letter. We have already observed that two words in (one for each case) in the period have one less letter than the others. The δ functions are introduced only to address this issue.
4. Future directions
We have handled the odd cases in two groups corresponding to residue classes modulo 4. We can also start with a uniform approach. For all odd let
It is clear that one can define a formula for L(N), which would be an analog of (Equation3.1(3.1) (3.1) ), using these values and the Dn’s. The period in this unified approach seems to be much larger than 11. Nevertheless, one intriguing claim analogous to Conjecture 3.2 is:
Conjecture 4.1.
Let n be an odd number , then or 3.
Using the last maximum absolute coefficients we can note that
Moreover, which is a closer estimate than Finch’s [Citation6] approximation for this value, but it neither supports nor disproves Kotesovec’s [Citation7] conjecture: (4.1) (4.1)
We also checked the possible fitting functions and their limits using our data to understand these constants better. We assume that these sequences look like for large n where the major contribution comes from the first two coefficients. Starting from n = 50000, we fitted the function to understand their asymptotic behavior better. We use the Aitken’s delta-squared series acceleration method [Citation9] that eliminates the a1 term in the asymptotics in estimating this limit. Then the approximation for the limiting constant a0 is
This estimated limit matches Kotesovec’s conjecture Equation(4.1)(4.1) (4.1) in 10 decimal digits.
Acknowledgment
The authors would like to sincerely thank the interest and encouragement of Jakob Ablinger, Stefano Capparelli, Steven Finch, Ralf Hemmecke, Veronika Pillwein, Carsten Schneider, Zafeirakis Zafeirakopoulos, and Wadim Zudilin. We are particularly grateful to Christoph Koutschan and Elaine Wong and the anonymous referee for their interest, careful reading, and helpful comments on the manuscript.
The authors would like to acknowledge Johannes Blümlein and Carsten Schneider for allowing us to use their servers for the initial calculations at the Research Institute for Symbolic Computation (RISC), Ralf Wahner and Werner Danielczyk-Landerl for their help in the storage of the early calculations at RISC, and Johann Messner from the MACH2 team for his help in carrying out the calculations.
Declaration of Interest
No potential conflict of interest was reported by the author(s).
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References
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