The Partition-Frequency Enumeration Matrix
aa r X i v : . [ m a t h . N T ] F e b THE PARTITION-FREQUENCY ENUMERATION MATRIX
HARTOSH SINGH BAL AND GAURAV BHATNAGAR
Abstract.
We develop a calculus that gives an elementary approach to enu-merate partition-like objects using an infinite upper-triangular number-theoreticmatrix. We call this matrix the Partition-Frequency Enumeration (PFE)matrix. This matrix unifies a large number of results connecting number-theoretic functions to partition-type functions. The calculus is extended toarbitrary generating functions, and functions with Weierstrass products. As aby-product, we recover (and extend) some well-known recurrence relations formany number-theoretic functions, including the sum of divisors function, Ra-manujan’s τ function, sums of squares and triangular numbers, and for ζ (2 n ),where n is a positive integer. These include classical results due to Euler, Ewell,Ramanujan, Lehmer and others. As one application, we embed Ramanujan’sfamous congruences p (5 n + 4) ≡ τ (5 n + 5) ≡ Introduction
Let σ ( n ) be the sum of divisors of n . Euler [16] showed that σ ( n ) = σ ( n −
1) + σ ( n − − σ ( n − − σ ( n −
7) + σ ( n −
12) + · · · . (1.1)There are two striking features of this result. One, this is a recurrence relationfor σ ( n ), a multiplicative arithmetic function. And two, the numbers appearing inEuler’s recurrence: 1, 2, 5, 7, . . . , are the generalized pentagonal numbers which arerelated to the partition function—the number of ways of writing a positive integeras an unordered sum of positive integers—which is part of additive number theory.The numbers are quite far apart, which makes it convenient to compute σ ( n ) forsmall values of n . These numbers feature in Euler’s pentagonal number theorem,which is the expansion(1 − q )(1 − q )(1 − q ) · · · = 1 − q − q + q + q − q − q + · · · . As is well-known, the reciprocal of the product appearing in Euler’s pentagonalnumber theorem is a generating function of integer partitions.Such recurrences have been found for many number-theoretic functions, includ-ing Ramanujan’s τ function and r k , the number of ways of writing a number n as anunordered sum of k squares; in addition, there are many recurrence relations thatcontain a partition function. Aside from Euler, such results have been found byRamanujan, Lehmer, Ewell; and many are found in number theory texts withoutattribution. Date : February 9, 2021.2010
Mathematics Subject Classification.
Primary: 11P81; Secondary: 11P83, 11A25, 05A17.
Key words and phrases.
Integer partitions, recurrence relations, divisor functions, sums oftriangular numbers, sums of squares, zeta function at even integers.
The objective of this paper is to study a number-theoretic matrix that seems tobe at the heart of the connection of arithmetic functions with partition functions,and such recurrence relations.Consider the matrix A = . . . . . . . . . . . . . . . ... ... ... ... ... ... ... , (1.2)with the ( i, j )th entry given by 1 when i | j , and 0 otherwise. Let A n be the n × n sub-matrix consisting of the entries from the first n rows and columns of A . For n = 1 , , . . . , in turn, we generate the vectors P n = ( P ( n − , P ( n − , . . . , P (0))and F ( n ) = ( F ( n ) , F ( n ) , . . . , F n ( n )) by means of the two equations A n P tn = F ( n ) t (1.3a) n X k =1 kF k ( n ) = nP ( n ) (1.3b)together with the initial condition P (0) = 1. (Here the superscript t is used todenote transpose.) The first few terms of the sequence P ( n ) generated in thisfashion are1 , , , , , , , , , , , , , , , , , , , , , . . .. This is the sequence p ( n ) of integer partitions. (See Example 2.1 for more details.)The objective of this paper is to utilize this elementary observation to find resultsthat connect the arithmetical functions of number theory with partition functions.We develop a calculus to determine the matrix associated with all functions thatcan be represented as infinite products or series. As a by-product, we recover manyclassical identities and recurrence relations, and find analogues for other interestingfunctions, and some natural generalizations.Before describing our results, as motivation, we prove our assertion that thesequence obtained from (1.3a) and (1.3b) is in fact p ( n ), the number of ways ofwriting n as an unordered sum of numbers.We find it convenient to use the symbols u = ( u , u , . . . ) to represent partitions.The symbol u k represents k , and ju k will represent k + k + · · · + k ( j times). Forexample, 3 u + u represents the partition 2 + 1 + 1 + 1 of 5. Thus we representpartitions by λ = X k f k u k , where f k ≡ f k ( λ ) are non-negative integers. This way of representing partitionsis not standard; however, it will come in handy as we extend our ideas to coloredpartitions and overpartitions.The symbols λ ⊢ n and | λ | = n are both used to say that λ is a partition of n .If | λ | = n , we have n = X k kf k . (1.4) HE PARTITION-FREQUENCY ENUMERATION MATRIX 3
The quantity f k ( λ ) denotes the frequency of k in λ , that is, the number of times k comes in λ . Let F k ( n ) = X λ ⊢ n f k ( λ )be the number of k ’s appearing in the partitions of n . By summing (1.4) over allpartitions of n , we have n X k =1 kF k ( n ) = np ( n ) . This shows that P ( n ) = p ( n ) satisfies (1.3b).To obtain (1.3a), observe that for k = 1 , , . . . , n , F k ( n ) = p ( n − k ) + F k ( n − k ) , (1.5)because adding u k to each partition of n − k yields a partition of n , and vice-versa,on deletion of u k from any partition containing k as a part, we obtain a partitionof n − k . This gives, on iteration, F k ( n ) = p ( n − k ) + p ( n − k ) + p ( n − k ) + · · · , for k = 1 , , . . . . (We take p ( m ) = 0 for m < P (0) = 1 deter-mine the sequence P ( n ) (as well as F k ( n ) for k = 1 , , . . . , n ). Since p ( n ) satisfiesthe equations and the initial condition, we must have P ( n ) = p ( n ). This provesour assertion earlier in the paper.We call such a matrix the Partition-Frequency Enumeration (PFE) matrix, sinceit contains all the information required to enumerate both partitions and the asso-ciated frequency function.As an immediate consequence of (1.3a) and (1.3b), we obtain a recurrence rela-tion that appears in Ramanujan’s work (see [11, p. 108]), but has been credited toFord [18]: n X d =1 σ ( d ) p ( n − d ) = np ( n ) . (1.6)This follows by multiplying both sides of (1.3a) by the diagonal matrixdiag(1 , , , . . . , n );taking column sums (to get each summand on the left hand side of (1.6)); and then,summing the column sums using (1.3b), to obtain the right hand side.Ramanujan’s recurrence highlights the relevance of the matrix to the connectionbetween arithmetical functions and partitions. There is nothing special about thearithmetic function σ ( n ), or indeed, the partition function. This kind of analysiscan be done for many different types of functions. In this paper, we develop a cal-culus for writing down matrices corresponding to many such partition-type objects,and illustrate their use in finding number-theoretic results. Further, we extend theideas to many other functions too, as the following sample of results attest.By essentially the same technique as (1.6), we find the recurrence ζ (2 n ) = ( − n +1 nπ n (2 n + 1)! + n − X k =1 ( − n − k +1 π k (2 k + 1)! ζ (2 n − k ) , (1.7) H. S. BAL AND G. BHATNAGAR where n is a positive integer, and ζ (2 n ) is given by the series ζ (2 n ) = ∞ X k =1 k n . This was discovered by Song [28] and given a proof using Fourier series. Ourderivation does not involve anything more than what Euler himself used to prove(1.7) for n = 1, that is, his evaluation ζ (2) = ∞ X k =1 k = π . As another example, consider Ramanujan’s recurrence for his τ function. Recallthe notation for q -rising factorials. For 0 < | q | <
1, we use( a ; q ) ∞ := ∞ Y k =0 (1 − aq k );and, ( a , a , . . . , a n ; q ) ∞ := ( a ; q ) ∞ ( a ; q ) ∞ · · · ( a n ; q ) ∞ . Ramanujan’s τ function is defined by the relation( q ; q ) ∞ = ∞ X n =0 τ ( n + 1) q n . Ramanujan’s [25, p. 152] recurrence for the τ function( n − τ ( n ) = ∞ X j =1 ( − j +1 (2 j + 1) ( n − − j ( j + 1) / τ (cid:0) n − j ( j + 1) / (cid:1) (1.8)looks very much like Euler’s recurrence (1.1); our proofs of the two results are quitesimilar, too.This paper is organized as follows. In § §
3, we show thatthe form of the matrix obtained is responsible for the relations between arithmeticalfunctions and partition-type objects. In § §
5, we consider a problem of Heninger, Rains and Sloane [21]that arose in the context of taking n th roots of theta functions, and find relatedresults. Next, in §
6, we extend a formula due to Lehmer [23] concerning powersof the η function. Our extension is to arbitrary power series, and is thus widelyapplicable, and gives several new results. As a third application (in § p (5 m + 4) ≡ τ (5 m + 5) ≡ , into an infinite family of congruences: P r (5 m + 4) ≡ , if r ≡ , HE PARTITION-FREQUENCY ENUMERATION MATRIX 5 where P r ( n ) are the coefficients of the power series expansion of 1 / ( q ; q ) r ∞ . Here r is a rational number, and a few special cases of our results overlap with congruencesfound by Chan and Wang [14]. More such congruences appear in § Enumeration of Partitions
In this section, we find the Partition-Frequency Enumeration (PFE) matrix forall “partition-type functions” with generating functions of the form Q ( z, q ) = ∞ Y k =1 − zq k ) b k = ∞ X n =0 P ( z, n ) q n . (2.1)Here ( b k ) is any sequence of complex numbers and | q | < | z | <
1. Forconvenience, we suppress the dependency on z , by writing Q ( q ) for Q ( z, q ), P ( n ) = P ( z, n ) and so on. We refer the reader to [4, 6, 7] for background information onthe theory of partitions.We first write the system of equations in more generality. Lemma 2.1 (An enumeration lemma) . Let A be an infinite matrix, and A n be thesub-matrix formed by taking the first n columns of A . Suppose that A n has a finitenumber (say, m n ) of non-zero rows. Let U k and V k be some sequences with V k = 0 for all k . Let P ( n ) , for n = 0 , , , . . . , and F ( n ) , F ( n ) , . . . , F m n ( n ) , be related asfollows: A n P ( n − P ( n − ... P (0) = F ( n ) F ( n ) ... F m n ( n ) , (2.2a) and m n X k =1 U k F k ( n ) = V n P ( n ) . (2.2b) Then, given the initial condition P (0) , the above equations determine P ( n ) , and F k ( n ) , for k = 1 , , . . . , n .Proof. The proof is immediate. If we know P (0), P (1), . . . , P ( n − F ( n ), F ( n ), . . . , F m n ( n ) from (2.2a); and then, we obtain P ( n ) from (2.2b). (cid:3) Example 2.1 (Enumeration of partitions) . We calculate the first few values of p ( n ) by this approach, using (1.3a) and (1.3b). First, we see that(1) ( p (0)) = (1) = ( F (1)) , so P (1) = 1. Next, for n = 2, (1.3a) gives (cid:18) (cid:19) (cid:18) p (1) p (0) (cid:19) = (cid:18) (cid:19) (cid:18) (cid:19) = (cid:18) (cid:19) = (cid:18) F (2) F (2) (cid:19) . This gives F (2) = 2 and F (2) = 1. Thus 2 P (2) = 1 · ·
1, or P (2) = 2. For n = 3, we have = = F (3) F (3) F (3) . From (1.3b), we have 3 p ( n ) = 1 · · · p (3) = 3. H. S. BAL AND G. BHATNAGAR
We wish to emphasize, that given the matrix A , and the initial condition p (0) =1, we can generate p ( n ) for n > F k ( n ) satisfying (1.3b). As will becomeapparent, the enumeration of many partition functions can be done just like this,using a matrix that is not very far from (1.2).In most of our work, we will consider the weights of Lemma 2.1 given by U n = V n = n and A n as n × n sub-matrices of an infinite matrix A . The matrix equationsare of the form A n P ( n − P ( n − P (0) = F ( n ) F ( n )... F n ( n ) , (2.3a)or, in weighted form, A ′ n P ( n − P ( n − P (0) = F ( n )2 F ( n )... nF n ( n ) . (2.3b)The entries of the matrix A n are denoted a i ( j ) for i, j = 1 , , . . . , n . The corre-sponding entries of A ′ n are then ia i ( j ).We require notations for the generating functions of the rows of the matrix A and the generalized frequency function. These are, respectively, R k ( q ) := X j a k ( j ) q j ; and N k ( q ) := X n ≥ F k ( n ) q n . Given a PFE matrix A , we refer to the corresponding sequences as the general-ized or the corresponding partition (respectively, frequency) functions, even if sucha combinatorial interpretation does not exist.Recall the notation for the rising factorials:( a ) = 1 , ( a ) r = a ( a + 1) · · · ( a + r −
1) for r > | z | < − z ) − a = ∞ X r =0 ( a ) r r ! z r . In view of the Binomial theorem, we see that the generalized partition function isgiven by P (0) = 1, and P ( n ) = X π = P riui | π | = n Y i ( b i ) r i r i ! z r i , (2.4)where the sum is over all partitions of n . Note that P i r i is the number of partsof the partition π , so the power of z keeps track of the number of parts. We definethe generalized frequency function F k ( n ) ≡ F k ( z, n ) as follows. F k ( n ) := X π = P riui | π | = n,rk> r k Y i ( b i ) r i r i ! z r i . (2.5) HE PARTITION-FREQUENCY ENUMERATION MATRIX 7
The sum in the definition of F k ( n ) is over all partitions π of n which contain k asa part. We now compute the associated PFE matrix. The following theorem givesus all the elements required by Lemma 2.1. Theorem 2.2.
Let | z | < , | q | < , and ( b k ) k ≥ be a sequence of complex numbers.Let Q ( q ) be the generating function (2.1) , and let P ( n ) and F k ( n ) the associatedfunctions given by (2.4) and (2.5) . Then:(a.) P (0) = 1 .(b.) The matrix equation (2.3a) holds, with entries of A n given by a i ( j ) = ( b i z r , if j = ri ;0 , otherwise . ; (2.6a) or, equivalently, the matrix equation (2.3b) holds, with entries of A ′ n given by a ′ i ( j ) = ( ib i z r , if j = ri ;0 , otherwise . . (2.6b) (c.) For n = 1 , , , . . . , n X k =1 kF k ( n ) = nP ( n ) . (2.7) Remark.
This work arose in the context of [9]; we wanted a convenient approachto enumerate objects considered in that paper.
Proof.
Clearly, from (2.1), we have P (0) = 1.For part (b.), we first prove a generalization of (1.5): for k = 1 , , . . . , F k ( n ) = zF k ( n − k ) + b k zP ( n − k ) . (2.8)To show this, we use the elementary identity r k ( b k ) r k r k ! = ( r k −
1) ( b k ) r k − ( r k − b k ( b k ) r k − ( r k − r k >
0. This gives F k ( n ) = X π = P riui | π | = n,rk> r k ( b k ) r k r k ! z r k Y i = k ( b i ) r i r i ! z r i = z X π = P riui | π | = n,rk> ( r k −
1) ( b k ) r k − ( r k − z r k − Y i = k ( b i ) r i r i ! z r i + b k z X π = P riui | π | = n,rk> ( b k ) r k − ( r k − z r k − Y i = k ( b i ) r i r i ! z r i = zF k ( n − k ) + b k zP ( n − k ) . This proves (2.8). Next, multiply both sides of (2.8) by q k and sum over k to obtain N k ( q ) = zq k N k ( q ) + b k zq k Q ( q ) , that is, N k ( q ) = b k zq k − zq k Q ( q ) . (2.9) H. S. BAL AND G. BHATNAGAR
This immediately gives an expression for the generating function of the rows of theassociated PFE matrix A : R k ( q ) = b k zq k − zq k , (2.10)Thus, for n ≥
1, we have the equations (2.3a), where the entries (2.6a) for thematrix are obtained by expanding the denominator of (2.10) as a geometric series,and comparing coefficients. To obtain the entries (2.6b) of the equivalent form(2.3b), we multiply both sides of (2.3a) by the vector (1 , , . . . , n ).Finally, (2.7) follows by multiplying (2.5) by k , and summing over k . The sumcan be interchanged for each partition π = P i r i u i of n and we use X i ir i = n to obtain the result. (cid:3) Examples 2.1.
A few examples of partition functions are given below. They canall be enumerated using the approach of Example 2.1.(1)
Partitions with distinct parts : Partitions with distinct parts are gen-erated by ( − q ; q ) ∞ ; that is, when z = −
1, and b k = − k in (2.1).Equation (2.8) reduces to F k ( n ) = P ( n − k ) − F k ( n − k ) , where P ( n ) = p ( n | distinct parts) in this context. Here we have used theself-explanatory notation from [7] for partitions with distinct parts. Thefirst row of the PFE matrix is[1 , − , , − , . . . ] . The remaining rows can be obtained from (1.2) by changing signs of thealternate non-zero entries too.(2)
Partitions with odd parts : The partitions with only odd parts are gen-erated by z = 1, b k = 0 and b k − = 1 for k > F k ( n ) are all 0 for all n .(3) Parts in S : If S is a subset of the natural numbers, and P ( n ) the partitionswith parts in S , the PFE matrix has entries a i ( j ) = ( , if i ∈ S and j = ri ;0 , otherwise . (4) Colored partitions : Colored partitions, with part k coming in b k colors,are given by the generating function (2.1) with z = 1 and b k ∈ N . Theyare also called prefabs; see Wilf [30, § Plane partitions : Plane partitions can be considered as colored partitionswhere the part k comes in k colors (see [7, Th. 15, p. 101]), that is, when b k = k for all k in (2.1).The next result gives the PFE matrix for a product of two generating functionsof the form (2.1). HE PARTITION-FREQUENCY ENUMERATION MATRIX 9
Theorem 2.3.
Consider the generating function ∞ Y k =1 − zq k ) b k ∞ Y k =1 − tq k ) c k = ∞ X k =1 P ( n ) q n . We can find the PFE matrix in the following two forms.(a.) A (1) n is an n × n matrix with entries given by a i ( j ) = ( b i z r + c i t r , if j = ri ;0 , otherwise . (2.11) (b.) A (2) n is a n × n matrix with entries given by a i ( j ) = b i z r , if i = 2 s − , j = rs for s = 1 , , . . . , n ; c i t r , if i = 2 s, j = rs for s = 1 , , . . . , n ;0 , otherwise . (2.12) Proof.
In this case the generalized partition function is given by a sum of the form P ( n ) = X ( π,µ ) Y i ( b i ) r i r i ! z r i Y j ( c j ) s j s j ! t s j , where the sum is over pairs of partitions ( π, µ ) with | π | + | µ | = n ; here π = P i r i u i and µ = P i s i v i . The parts of π are accounted by powers of z and parts of µ bypowers of t . The corresponding generalized frequency functions can be defined asin (2.5). They satisfy F zk ( n ) = zF zk ( n − k ) + b k zP ( n − k ); F tk ( n ) = tF tk ( n − k ) + c k tP ( n − k ) . This implies the matrix equation implied by (2.12). In addition, adding the twoequations we obtain F zk ( n ) + F tk ( n ) = zF zk ( n ) + tF tk ( n ) + ( b k z + c k t ) P ( n − k )= ( b k z + c k t ) P ( n − k ) + ( b k z + c k t ) P ( n − k ) + · · · . This implies the entries given in (2.11). (cid:3)
Remark.
Note that the implied matrix equations have two different columns onthe right hand side of (2.3a) or (2.3b). The choice between the two forms (2.11)and (2.12) is dictated by whether we are interested in computing the frequencyfunctions F zk and F tk ( n ) individually. In either case, the result analogous to (2.7) isthe same. We have n X k =1 k (cid:0) F zk ( n ) + F tk ( n ) (cid:1) = nP ( n ) . (2.13)As a prototypical example, we consider the overpartitions introduced by Corteeland Lovejoy [15]. Example 2.2 (Overpartitions) . An overpartition of n is a non-increasing sequenceof natural numbers whose sum is n , where the first occurrence of a number can be overlined. We denote the number of overpartitions of n by p ( n ). The generatingfunction of overpartitions is the product of the respective generating functions: Q ( q ) = X n ≥ p ( n ) q n = ( − q ; q ) ∞ ( q ; q ) ∞ . We obtain the PFE matrix from (2.12) by taking z = − b j = − t = 1 and c j = 1. The matrix consists of rows of the PFE matrix corresponding to distinctpartitions alternating with the PFE matrix for ordinary partitions.We have seen that the enumeration of many types of partitions can be enumer-ated using PFE matrices very similar to (1.2). As a bonus, we obtain informationabout the frequency function associated with partitions. Next, we explore the con-nection to arithmetical functions; this is an immediate consequence of the form ofthe PFE matrix. 3. Connection with arithmetic functions
The objective of this section is to illustrate the connection between the arith-metical functions of number theory and partition functions. The connection is dueto the following proposition, which applies whenever we have a PFE matrix withentries a mild variation of the entries of the matrix (1.2).
Proposition 3.1.
Let P ( n ) and F k ( n ) satisfy P (0) = 1 , the matrix equations (2.3a) with entries a i ( j ) given by (2.6a) (or equivalently, (2.3b) with entries ia i ( j ) ),and (2.7) . Let f : N → C be a function and let g ( n ) be defined as g ( n ) := X d | n b d f ( d ) z n/d . (3.1) Then we have the following: n X k =1 g ( k ) P ( n − k ) = n X k =1 f ( k ) F k ( n ); (3.2a) Q ( q ) ∞ X k =1 g ( k ) q k = ∞ X n =0 (cid:16) n X k =1 f ( k ) F k ( n ) (cid:17) q n ; (3.2b) Remarks. (1) When f ( k ) = k , then in view of (2.7), equations (3.2a) and (3.2b) reduceto n X k =1 g ( k ) P ( n − k ) = nP ( n ); (3.3a) Q ( q ) ∞ X k =1 g ( k ) q k = ∞ X n =0 nP ( n ) q n . (3.3b)(2) When z = 1, and g ( n ) and b k f ( k ) are both arithmetic functions, then weobtain results connecting number-theoretic functions to partition functions.(3) The sequence g ( n ) is the sequence of the weighted sums of the n th columnof the matrix given by in (2.6a); more precisely, g ( n ) is the n th column sumof the matrix ( f (1) , f (2) , . . . , f ( n )) A n . HE PARTITION-FREQUENCY ENUMERATION MATRIX 11 (4) Many results that fit the form of Proposition 3.1 were previously obtainedby a standard technique, of taking logarithmic derivatives of the relevantgenerating functions. Thus the enumeration result in Theorem 2.2 can beconsidered an alternate approach to this standard technique.
Proof.
To obtain (3.2a), multiply the k th row by f ( k ) on both sides of the equation(2.3a). The k th row on either side is X j> f ( k ) a k ( j ) P ( n − j ) = f ( k ) F k ( n ) , where a i ( j ) is given by (2.6a). Now summing over k , and noting that a k ( j ) = b k z j/k when k | j and 0 otherwise, we obtain the result.To obtain (3.2b), we multiply both sides of (3.2a) by q n and sum over n . (cid:3) Let P ( n ) and Q ( q ) = Q ( z, q ) be given by (2.1). By Theorem 2.2, the conditionsof the proposition are satisfied, and we have (3.2a) and(3.2b). In the examplesbelow, we illustrate the application of these to obtain several classical and someresults results. In addition, we use (2.9) in the form: b k zq k − zq k Q ( q ) = ∞ X n =0 F k ( n ) q n . (3.4) Example 3.1 (Variations of Ramanujan’s recursion) . Ramanujan’s recurrence(1.6) follows from (3.2a). Let z = 1, b k = 1 and f ( k ) = k , for all k , so that g ( n ) = σ ( n ). There are several ways one can find analogous results. Let σ k ( n ) nowdenote the sum of k th powers of the divisors of n (with σ ( n ) = σ ( n )). In all thefollowing, we take z = 1.First take b k = k and f ( k ) = k . Then g ( n ) = σ ( n ), the sum of squares ofthe divisors of n . We obtain a well-known recursion for plane partitions (denoted P L ( n )) nP L ( n ) = n X d =1 σ ( d ) P L ( n − d ) . See, for example, [24, Ex. 7, pg. 28]. Alternatively, let b k = r , so we obtain coloredpartitions where each part k comes in r colors. In this case, we obtain np r ( n ) = r n X d =1 σ ( d ) p r ( n − d ) , where we have denoted the number of associated partitions by p r ( n ). This gives asolution to Apostol [8, Ex. 9, Ch. 14]. (However, our solution is valid even when r is not a positive integer.)Another possibility is to take f ( k ) = k m (keeping b k = 1) for all k . We obtainan expression for the m th moment of the frequency function corresponding to theordinary partitions. M m ( n ) := n X k =1 k m F k ( n ) = n X d =1 σ m ( d ) p ( n − d ) . (3.5) Finally, another variation is obtained by taking different arithmetic functions.For example, consider the µ function defined as µ ( n ) := , if n = 1;( − k , if n = p . . . p k ;0 , otherwise. (3.6)It is well known [8, p. 25] that X d | n µ ( d ) = ( , if n = 10 , otherwise . (3.7)Taking the case b k = 1, f ( k ) = µ ( k ) gives n X k =1 µ ( k ) F k ( n ) = p ( n − , (3.8)which indicates the close relationship of F k ( n ) with the partition function. Anal-ogous results can be obtained for b k = r (a constant), and, when f ( k ) is anyother interesting arithmetic function (such as Euler’s totient function or Liouville’sfunction), where there are identities of the form g ( n ) = X d | n f ( d ) . Example 3.2. ([8, p. 327]) We now obtain a cute result regarding the connection of µ ( n ) with the exponential function. We want to use (3.7) to obtain a nice expressionfor g ( n ). Take z = 1, b k = µ ( k ) /k and f ( k ) = k in (3.1) to obtain g ( n ) = ( , if n = 10 , otherwise . Then (3.3a) reduces to nP ( n ) = n X k =1 g ( k ) P ( n − k ) = P ( n − . This gives, on iteration, P ( n ) = 1 /n !, and Q ( q ) = ∞ X n =0 q n n ! = e q ;or, ∞ Y k =1 (cid:0) − q k (cid:1) − µ ( k ) k = e q . This is valid for | q | <
1, and as formal power series.The next example uses a technique used by Euler. We use Euler’s pentagonalnumber theorem: ( q ; q ) ∞ = ∞ X k = −∞ ( − k q k (3 k − = ∞ X n =0 e ( n ) q n , (3.9) HE PARTITION-FREQUENCY ENUMERATION MATRIX 13 where e ( n ) = ( ( − k , if n = k (3 k ± ;0 , otherwise. Example 3.3. (Moments of the frequency function) We extend some formulas ofAndrews and Mirca [2]. Take b k = 1 and f ( k ) = k m for all k . The (3.2b) can bewritten as ∞ X n =0 M m ( n ) q n = ∞ X n =0 n X k =1 k m F k ( n ) q n = 1( q ; q ) ∞ ∞ X n =0 σ m ( n ) q n , where we have used M m ( n ) to denote the m th moment of the frequency functioncorresponding to partitions. Multiplying both sides by the product ( q ; q ) ∞ , using(3.9) and comparing coefficients of q n on both sides, we obtain σ m ( n ) = ∞ X j = −∞ ( − j M m ( n − j (3 j − / . (3.10)For m = 0 and m = 1, we obtain results of Andrews and Merca [2].A related result is obtained by considering (3.4) in the above case (with m = 0).We find that ∞ X n =1 q kn = ( q ; q ) ∞ ∞ X n =0 F k ( n ) q n . On comparing the coefficients of q n on both sides we find that ∞ X j = −∞ ( − j F k ( n − j (3 j − /
2) = ( , if k | n ;0 , otherwise. (3.11)This is a refinement of the last formula of [2]. Evidently P k F k ( n ) is the totalnumber of parts occurring in all the partitions of n . If we sum over k , we obtaintheir formula.Next we consider Euler’s recurrence (1.1). What is different from the previousexample is that we consider different partition functions. Example 3.4 (Recurrences for σ ( n )) . To obtain (1.1), we take b k = − k .The “partitions” are given by e ( n ) in Euler’s formula (3.9). We take f ( k ) = k , sothat g ( n ) = − σ ( n ). Now (3.3b) and (3.9) give( q ; q ) ∞ ∞ X n =1 ( − σ ( n ) q n = (cid:18) ∞ X k = −∞ ( − k q k (3 k − (cid:19)(cid:18) ∞ X n =0 σ ( n ) q n (cid:19) = ∞ X n =0 ne ( n ) q n . On comparing coefficients of q n we obtain (1.1) in the form ∞ X j = −∞ ( − j +1 σ (cid:0) n − j (3 j − / (cid:1) = ne ( n ) . (3.12)(To match the two forms, we take σ (0) = n (if it occurs) in (1.1).)To introduce a variation, we now use Jacobi’s result [6, p. 500]:( q ; q ) ∞ = ∞ X k =0 ( − k (2 k + 1) q k ( k +1)2 . (3.13) Now we take b k = − f ( k ) = k . Here Q ( q ) reduces to ( q ; q ) ∞ , and we use (3.13)rather than (3.9) to expand the products. On comparing coefficients as before, weobtain a result of Ewell [17, Th. 3]: ∞ X k =0 ( − k (2 k + 1) σ (cid:0) n − k ( k + 1) / (cid:1) = ( ( − k +1 k ( k +1)(2 k +1)6 , if n = k ( k +1)2 ;0 , otherwise . Another interesting identity is obtained by noting that( q ; q ) ∞ ∞ X n =0 σ ( n ) q n = ∞ X n =0 ne ( n ) q n , and, − q ; q ) ∞ ∞ X n =0 σ ( n ) q n = ∞ X n =0 np ( n ) q n . On multiplying the two and taking the coefficient of q n on both sides, we obtainan identity for the convolution of the σ function in terms of the partition function n − X k =1 σ ( k ) σ ( n − k )= ∞ X j = −∞ ( − j +1 (cid:0) n − j (3 j − / (cid:1)(cid:0) j (3 j − / (cid:1) p (cid:0) n − j (3 j − / (cid:1) . (3.14)For another recurrence relation for σ ( n ), see Example 4.5.We have now seen several techniques to obtain results connecting the arithmeticalfunctions of elementary number theory with partition-type objects. Next, we extendthe applicability of these techniques by considering arbitrary generating functions.4. A calculus for obtaining PFE Matrices
We have seen in Section 2 that given the PFE matrix, we can determine P ( n )and F k ( n ). In addition, we took P ( n ) to be generated by infinite products of theform (2.1), or their products, and computed the matrix. We now consider anypower series of the form Q ( q ) = ∞ X n =1 P ( n ) q n (4.1)and find a PFE matrix associated with it. This will enhance the applicability ofthe techniques of Section 3. Theorem 4.1.
Let P ( n ) be a given sequence with P (0) = 1 . Then there is a uniquesequence ( b n ) , and a frequency function F k ( n ) , for k = 1 , , . . . , n , satisfying (2.7) and matrix equations of the form (2.3b) , where the entries of the sequence of n × n matrices A ′ n are given by a i ( j ) = ( ib i , if i | j ;0 , otherwise . Remarks.
HE PARTITION-FREQUENCY ENUMERATION MATRIX 15 (1) The P ( n ) in this theorem may depend on z too, and b j may depend on z . Thus the matrix obtained may be different from that obtained in Theo-rem 2.2.(2) Proposition 3.1 (with z = 1) applies to the case where Q ( q ) is given as apower series.(3) Theorem 2.3 applies to the case where the generating function is given bya product of series too. Proof.
The proof is by induction. Note that (2.7) for n = 1 gives F (1) = p (1).Now the entries of the first row of A n (which are all equal to a (1) = b ) can bedetermined from (2.3b). Next if b , b , . . . , b n − are known, (2.3b) gives the values F ( n ), F ( n ), . . . , F n − ( n ). Use (2.7) to determine F n ( n ), and then (2.3b) againto find b n . (cid:3) We illustrate the use of this theorem by considering an example from Andrews[3, Problem 12-2] (see also [5, p. 105], [7, p. 31] and [13]).
Example 4.1 (How to discover the (first) Rogers–Ramanujan identity) . Consider P ( n ) to be the set of partitions of n with the property that the difference betweenparts is at-least 2. The first few values, for n = 0 , , , . . . of P ( n ), are: 1, 1, 1,1, 2, 2, 3, 3, 4, 5, 6, 7, 9, 10, 12, 14. From here we compute the matrix usingTheorem 4.1.We find that F (1) = 1. This gives b = 1. Next, for n = 2, (1.3a) gives (cid:18) b (cid:19) (cid:18) (cid:19) = (cid:18) b (cid:19) = (cid:18) F (2)2 F (2) (cid:19) . This gives F (2) = 2. Now from (2.7), we get and 2 + 2 F (2) = 2, so 2 F (2) =2 b = 0. Thus, b = 0. For n = 3, we have b = F (3) . This gives 3 p ( n ) = 1 · · · F (3) = 3, so F (3) = 0 = b . The next stepgives b = b . This gives F (4) = 4, F (4) = F (4) = 0 and 4 + 4 b = 4 ·
2, which gives b = 1.Carrying on this this fashion, we can discover that the first few values of b k aregiven by the sequence b k = ( k ≡ ∞ Y k =0 − q k +1 )(1 − q k +4 ) , which generates partitions with parts congruent to 1 or 4 (mod 5). This shows howone can discover the (first) Rogers–Ramanujan identity.This example shows that we can use Theorem 4.1 to experimentally obtain aninfinite product of the form1 + ∞ X k =1 P ( n ) q n = ∞ Y k =1 (cid:0) − q k (cid:1) − b k (4.2)corresponding to a power series. Andrews [5, Theorem 10.3] has given an algorithm(usually called Euler’s algorithm) for this purpose. The relationship of Theorem 4.1with Andrews’ approach will become clear with the following theorem, which re-verses the process we have followed so far. Given an arithmetical sequence g ( n ), wefind an associated PFE matrix together with the associated partition and frequencyfunctions. Theorem 4.2.
Let g ( n ) , n = 1 , , , . . . , be any sequence. Then there are sequences P ( n ) , for n = 1 , , . . . , with P (0) = 1 ; F k ( n ) for k = 1 , , . . . , n ; and a sequence b n , satisfying (2.7) and the matrix equation (2.3b) , with entries given by a ′ i ( j ) = ( ib i , if j = ri ;0 , otherwise ; ; such that, nP ( n ) = n X k =1 g ( k ) P ( n − k ) . (4.3) In addition, the sequences b n , P ( n ) and F k ( n ) are uniquely determined.Proof. We define b n by means of the equation g ( n ) = X d | n db d ; (4.4a)so, by M¨obius inversion [8, p. 30], we have nb n = X d | n µ (cid:0) nd (cid:1) g ( d ) , (4.4b)where µ ( n ) is defined in (3.6). Next take Q ( q ) as in (2.1) with z = 1 to obtain theassociated partition function P ( n ) and the associated frequency function F k ( n ).Theorem 2.2 implies they satisfy the matrix equation (2.3b) with entries as given,and (2.7). Further, Proposition 3.1 (with z = 1) implies (4.3). Finally, note that(4.4b) determines b n , and by Lemma 2.1, the sequences P ( n ), F k ( n ) are uniquelydetermined. (cid:3) The aforementioned algorithm of Andrews [5, Theorem 10.3] for finding theproduct form in (4.2) combines (4.3) and (4.4a) in the following two forms nP ( n ) = nb n + X d | nd One difference between Andrews’ approach and this paper is that we do not restrictthe use of Euler’s algorithm to the case where P ( n ) and b k in (4.2) are integers.Next, we give a few examples to illustrate a calculus for finding the PFE matrixfor various kinds of functions. First we show how one can find the PFE matrix forfinite products. Example 4.2 (The complete symmetric function) . We compute the PFE matrixfor the complete homogeneous symmetric polynomial h k ( x ) in m variables x =( x , . . . , x m ). Take Q m ( q ) = m Y k =1 − x k q = ∞ X n =0 h n ( x ) q n . For fixed k , the PFE matrix for 1 / (1 − x k q ) is the row matrix (cid:2) x k , x k , x k , . . . (cid:3) . This follows from Theorem 2.2 with z x k . In view of Theorem 2.3, applied inthe form (2.12), we obtain for any fixed m , and n = 1 , , , . . . an m × n matrix A mn with ( i, j )th entry x ji .The weighted sums g ( n ) from (3.1) reduce to p m ( x ), the power symmetric func-tions, and, in view of (2.13), equation (3.2a) reduces to [24, eq. 2.11]: n X k =1 p k ( x ) h n − k ( x ) = nh n ( x ) . In this example, we can increase the number of variables in x indefinitely, as isroutinely done in the theory of symmetric functions.There is another way to take a limiting process, which we illustrate by thefollowing example. Example 4.3 (A recurrence for ζ (2 n )) . We compute the PFE matrix ofsin πzπz = ∞ Y k =1 (cid:16) − z k (cid:17) = ∞ X k =0 ( − k π k z k (2 k + 1)!in two different ways.First note that for any fixed k , the PFE matrix for (1 − z /k ) is the row matrix (cid:20) − k , − k , − k , . . . (cid:21) , where the variable q is taken to be z . This follows from Theorem 2.2 with z /k , q z . Now let Q m ( z ) = m Y k =1 (cid:16) − z k (cid:17) = X n P mn z n . In view of Theorem 2.3, applied in the form (2.12), we obtain, for any fixed m , and n = 1 , , , . . . , the m × n matrix A mn with ( i, j )th entry a i ( j ) = − i j . This set of matrices can be used to compute P mn using Lemma 2.1 (take A n A mn , U k = 1 , V k = k, m n = m ). The function defined by (3.1) is given by g m ( n ) = ( − m X k =1 k n , and from (2.13) and (3.2a) we obtain n X k =1 g m ( k ) P mn ( n − k ) = nP mn ( n ) . As m → ∞ , this becomes a formula due to Song [28]: n X k =1 ( − ζ (2 m ) a n − m ) = na n , where a n = ( − n π n (2 n + 1)! . The coefficients a n are obtained from the coefficients of the power series forsin πz/πz . We can rewrite this formula in the form (1.7) mentioned in the in-troduction.On the other hand, taking g ( n ) = − ζ (2 n ), we can obtain a product of the form ∞ Y k =1 (cid:0) − z k (cid:1) − b k , where, from (4.4b), kb k = − X d | k µ (cid:0) k/d (cid:1) ζ (2 d ) . Thus we must have sin πzπz = ∞ Y k =1 (cid:0) − z k (cid:1) k P d | k µ ( k/d ) ζ (2 d ) . (4.6)This formula is valid as a formal power series in z . As an analytic formula, somefurther justification is required.We can extend this idea and write the PFE matrix of any function which canbe written as a Weierstrass product (see [1, Th. 7, p. 194]). This requires us tocalculate a PFE matrix for the exponentials of polynomials.We require a simple trick. Note that if g ( n ) is a given function, then one PFEmatrix that corresponds to it is the row matrix (cid:2) g (1) , g (2) , g (3) , . . . (cid:3) . This is easy to verify. However, this may not be the same matrix as generated byTheorem 4.2.We re-look at Example 3.2 to obtain a PFE matrix for e aq . Note that thecoefficients P ( n ) in the power series expansion of e aq satisfy nP ( n ) = aP ( n − . HE PARTITION-FREQUENCY ENUMERATION MATRIX 19 Comparing with (4.3), we see that g ( n ) defined as follows works: g ( n ) = ( a, for n = 1;0 , for n > . So one PFE matrix generating e aq is (cid:2) a, , , . . . (cid:3) . Similarly, one can generate a matrix for e aq m by taking a row matrix with a inthe m th place and 0s elsewhere. To obtain the matrix for the exponential of apolynomial in q , we use Theorem 2.3. Example 4.4 (Gamma Function) . The Gamma function has a Weierstrass productgiven by 1Γ( z ) = ze γz ∞ Y k =1 (cid:16) zk (cid:17) e − z/n , where γ is the Euler-Mascheroni constant.We consider Q m ( z ) = m Y k =1 (cid:16) zk (cid:17) e − z/k . We take the variable z in place of q . Note that for any fixed k , the PFE matrix for(1 + z/k ) e − z/k is the row matrix (cid:20) , − k , k , − k , . . . (cid:21) . Here we have added − /k in the first column to account for the factor e − z/k . Inview of Theorem 2.3, applied in the form (2.12), we obtain for any fixed m , and n = 1 , , , . . . and m × n matrix A mn with ( i, j )th entry a i ( j ) = ( , if j = 1;( − j − i j , for j > . The corresponding function defined in (3.1) is given by g m ( n ) = ( , if n = 1;( − n − P mk =1 1 k n , for n > . As in Example 4.3, we can take limits as m → ∞ . In addition, we can obtain analternate product formulation for 1 z Γ( z ) e γz by applying Theorem 4.2 to g ( n ) = ( , if n = 1;( − n − ζ (2 n ) , for n > . Example 4.5 (Jacobi Triple Product) . We now consider the product side of theJacobi Triple Product identity [12, p. 10]:1 + ∞ X n =1 (cid:0) z n + z − n (cid:1) q n = (cid:0) q , − zq, − q/z ; q (cid:1) ∞ . Here P ( n ) is given by P ( n ) = ( z m + z − m , if n = m ;0 , otherwise . We can compute the PFE matrix from the product side. A PFE matrix corre-sponding to the product (cid:0) q ; q (cid:1) ∞ has the ( i, j ) entry given by ( − , if i is even, and j = ri for some r ;0 , otherwise . A PFE matrix corresponding to the product (cid:0) − qz, − q/z ; q (cid:1) ∞ has entries ( ( − r − (cid:0) z r + z − r (cid:1) , if i is odd, and j = ri for some r ;0 , otherwise . Thus the PFE matrix of the product (cid:0) q , − zq, − q/z ; q (cid:1) ∞ is given by a i ( j ) = ( − r − (cid:0) z r + z − r (cid:1) , if i is odd, and j = ri for some r ; − , if i is even, and j = ri for some r ;0 , otherwise . We now apply this for the special case z = 1 in Jacobi’s triple product identity,which corresponds to Gauss’ identity [12, Eq. (1.3.13)]: ϕ ( q ) = ∞ X k = −∞ q k = 1 + 2 ∞ X k =1 q k = (cid:0) q ; q (cid:1) ∞ (cid:0) − q ; q (cid:1) ∞ . (4.7)Here the corresponding partition function is given by P ( n ) = , for n = 1;2 , for n = k , for some k ;0 , otherwise.Take f ( k ) = k to find that g ( n ) (in (3.1)) is given by g ( n ) = ( σ (2 m − , for n = 2 m − , m = 1 , , . . . ; − (cid:0) σ o (2 m ) + σ e (2 m ) , for n = 2 m, m = 1 , , . . . ;where σ o ( n ) (respectively, σ e ( n )) denotes sum of odd divisors (respectively, evendivisors) of n . Using the elementary observations σ o (2 m ) = σ (2 m ) − σ e (2 m ) , and, σ e (2 m ) = 2 σ ( m ) , we find that g ( n ) = ( σ (2 m − , for n = 2 m − , m = 1 , , . . . ; − σ (2 m ) + 2 σ ( m ) , for n = 2 m, m = 1 , , . . . . With Q ( q ) = ϕ ( q ), and g ( n ), P ( n ) as given here, from (3.3b), we obtain (cid:16) ∞ X n =1 q n (cid:17) ∞ X k =1 g ( k ) q k = 2 ∞ X j =0 j q j . HE PARTITION-FREQUENCY ENUMERATION MATRIX 21 For k > 0, we compare coefficients of q k to obtain12 g ( k ) + ∞ X j =1 g ( k − j ) = ( k, if k = j for some j ;0 , otherwise . Explicitly, this is the following pair of recurrence relations for σ ( k ) when k = 2 m − k = 2 m . σ (2 m − 1) + (cid:0) − σ (2 m − 2) + 2 σ ( m − (cid:1) + 2 σ (2 m − 5) + · · · = ( m − , if 2 m − j for some j ;0 , otherwise; (4.8a)and, σ (2 m ) − σ ( m ) − σ (2 m − 1) + (cid:0) σ (2 m − − σ ( m − (cid:1) + · · · = ( − m, if 2 m = j for some j ;0 , otherwise . (4.8b) Remark. Define the theta product as θ ( z ; q ) := ( z, q/z ; q ) ∞ . The approach of Example 4.5 can be used to find the PFE matrix for θ ( z/a ; q ) andits reciprocal, and thus, for rational products of factors of the form θ ( z/a ; q ); inother words, for a (multiplicative) elliptic function (see Rosengren [26, Th. 4.12]).In this section, we developed a calculus to obtain the entries of the PFE matrixfor a wide variety of functions. We now consider some applications of these ideas.5. Application 1: Roots of generating functions Let P ( n ) and b k be the corresponding sequences that satisfy (4.2). In this sectionwe consider the relationship between the divisibility properties of the two sequences P ( n ) and b k . As a by-product, we obtain a result due to Heninger, Rains and Sloane[21] which they obtained in the context of finding out n th roots of theta functions.We first prove a preliminary proposition about P ( n ) and b k . Proposition 5.1. Let P ( n ) and b k satisfy (4.2) . The P ( n ) is an integer (or arational number) if and only if b k is an integer (respectively, rational number).Proof. Let b k be an integer for all k . Then the P ( n ) are all integers too. Thisfollows from (2.4) and the fact that ( b ) r /r ! is an integer if b is an integer and r anon-negative integer.Conversely, suppose P ( n ) is an integer for all n . Let F k ( n ) be the correspondingfrequency function. We show all b k are integers. For the sake of contradiction, let m be the smallest number such that b m is not an integer. Clearly, in view of thematrix equation (2.3a), F k ( m ), for k = 1 , , . . . , m − P ( k )and b k are integers for k = 1 , , . . . , m − 1. Consider the product Q m − ( q ) = m − Y k =1 (cid:0) − q k (cid:1) − b k = ∞ X n =0 P ′ ( n ) q n . Let F ′ k ( n ) be the corresponding frequency function. Clearly, F ′ m ( m ) = 0, and from(2.7) we have mP ′ ( m ) = m − X k =1 kF ′ k ( m ) . From the equation (2.3a) for n = m , we must have F ′ k ( m ) = F k ( m ) for k =1 , , . . . , m − 1. Thus (2.7) implies that mP ( m ) = mF m ( m ) + m − X k =1 kF k ( m )= mF m ( m ) + mP ′ ( m ) . This implies that F m ( m ) is an integer, and since b m P (0) = F m ( m ), b m must be aninteger. This is a contradiction to our assumption that b m is not an integer.The proof of the result when P ( n ) and b k are rational numbers is straightforward. (cid:3) Next, we prove the main theorem of this section concerning divisibility propertiesshared by P ( n ) and b k . Theorem 5.2. Let p be a prime and r > a positive integer such that p r | P ( n ) for all n > . Then,(a) if ( p, m ) = 1 , then p r | b m ; and,(b) if p | m , then p r − | b m .Remark. If we take P ( n ) to be the sequence (1 , , , , . . . ), then a short calculationshows that b = 4 , b = − . So we have an example where 2 | P ( n ) for all n > | b , but 2 ∤ b . Proof. First consider the case ( p, m ) = 1. The proof is by induction on m . It iseasy to see that b = P (1), so the result holds for m = 1. From (4.5b), we have mb m = mP ( m ) − X d | md Clearly, each term on the right is divisible by p r , so p r | pb p or p r − | b p . Thiscompletes the proof for m = p and for part (b) of the theorem. (cid:3) Recall the notations (6.1) and (4.2). Corollary 5.3. Let P ( n ) with P (0) = 1 be as in (4.2) . Suppose for n > , P ( n ) are multiples of a fixed integer m t , then the formal series given by the m s th rootsof Q ( q ) will have integer coefficients for all s < t .Remark. This is another way to state a result of Heninger, Rains and Sloane [21,Cor. 2]. Proof. In the notation (6.1), we are interested in the coefficients P r ( n ) with r =1 /m s . Let b k correspond to the given sequence P ( n ) in (4.2). By Theorem 5.2, b k /m s are all integers since s ≤ t − 1. This implies the P r ( n ) are also integers,where r = 1 /m s , for all s < t . (cid:3) In this section we considered the n th roots of generating functions. In the nextsection, we consider arbitrary powers of generating functions.6. Application 2: Powers of generating functions The objective of this section is to extend a result due to Lehmer [23] (see (6.4)below) concerning powers of the eta function to arbitrary generating functions. Asa result we obtain analogous results for any generating function.We consider powers of Q ( q ) and use the notation P r ( n ) defined by Q ( q ) r = ∞ X n =0 P r ( n ) q n . (6.1)Here Q ( q ) r , for complex r , is considered a formal power series. We have the follow-ing theorem. Theorem 6.1. Let r and s be non-zero complex numbers and P r ( n ) and P s ( n ) bedefined by (6.1) . Then we have the recurrence relation: n X j =0 ( n − ( r/s + 1) j ) P r ( n − j ) P s ( j ) = 0 . (6.2) Proof. By Theorem 4.1 the hypothesis of Proposition 3.1 holds with z = 1. Wetake f ( k ) = k and b k = r, s and use (3.3b) to find that rQ ( q ) r ∞ X k =1 g ( k ) q n = ∞ X n =0 nP r ( n ) q n sQ ( q ) s ∞ X k =1 g ( k ) q n = ∞ X n =0 nP s ( n ) q n , where g ( n ) is as defined in (3.1). This gives rs (cid:16) ∞ X n =0 P r ( n ) q n (cid:17)(cid:16) ∞ X n =0 nP s ( n ) q n (cid:17) = (cid:16) ∞ X n =0 nP r ( n ) q n (cid:17)(cid:16) ∞ X n =0 P s ( n ) q n (cid:17) . The recurrence (6.2) follows by comparing coefficients of q n on both sides. (cid:3) Some examples of results following from Theorem 6.1 are as follows. Example 6.1 (Convolutions of Fibonacci Numbers) . Let Fib( n ) denote the Fi-bonacci numbers. The generating function is given by Q ( q ) = 11 − q − q = ∞ X n =0 Fib( n ) q n . We take s = − r ( n ) denote the sequence generated by Q ( q ) r . Then wehave the three-term recursion n Fib r ( n ) = ( n + r − 1) Fib r ( n − 1) + ( n + 2( r − r ( n − . The initial cases Fib r (0) = 1, Fib r (1) = r can be used to compute the sequences.Special cases of Fib r ( n ) for small positive integral values of r have appeared incombinatorial contexts, see OEIS [27, A001628, A001629, A001872-5]. At the timeof writing this paper, a three-term recurrence does not seem to be known for anyof these. Note that r need not be a positive integer in our formula. Example 6.2 (Powers of the η function) . Consider Q ( q ) = 1( q ; q ) ∞ . When a q -expansion is known for one value of s , it can be used to find a recurrencerelation for P r ( n ), where P r ( n ) is defined as in (6.1). Take s = − r ∞ X j = −∞ ( − j ( n + ( r − j (3 j − / P r (cid:0) n − j (3 j − / (cid:1) = 0 . (6.3)When r is negative, this gives a recurrence for the coefficients of the powers of the η function defined as η ( q ) := q / ( q ; q ) ∞ , where q = e πiτ for ℑ ( τ ) > 0. When r = − P r ( n ) = τ ( n ), and we obtain aresult used by Lehmer [22] to compute values of τ ( n ).Instead of (3.9), we use (3.13) and Theorem 6.1 with s = − ∞ X j =0 ( − j (2 j + 1) ( n + ( r/ − j ( j + 1) / P r (cid:0) n − j ( j + 1) / (cid:1) = 0 . (6.4)This is an extension of Ramanujan’s recurrence for τ ( n ) highlighted in the intro-duction (1.8). Example 6.3 (Sums of squares and triangular numbers) . In the examples so far,we have used theta function identities which are powers of ( q ; q ) ∞ to derive re-currence relations. In view of Theorem 4.1, we can apply the same approach toother generating functions. In fact, we don’t need the product form to obtain suchrecurrences. We use Ramanujan’s notation [12, p. 7]; let ϕ ( q ) = ∞ X k = −∞ q k = 1 + 2 ∞ X k =1 q k ; and, ψ ( q ) = ∞ X k =0 q k ( k +1)2 . Let r k ( n ) and t k ( n ) be defined by ϕ ( q ) k = ∞ X n =0 r k ( n ) q n and ψ ( q ) k = ∞ X n =0 t k ( n ) q n . HE PARTITION-FREQUENCY ENUMERATION MATRIX 25 Then r k ( n ) is the number of ways of writing n as an ordered sum of k integers and t k ( n ) the number of ways of writing n as an ordered sum of k triangular numbers.(Here order matters: 1 + 6 = 6 + 1 are considered different.)We take s = 1 and Q ( q ) = ϕ ( q ) in Theorem 6.1. Note that r ( n ) = , if n = 1;2 , if n = j for some j ;0 , otherwise . Using this we find an identity in which is proved using Liouville’s methods inWilliams [31, p. 77] (see also Venkov [29, p. 204]): nr k ( n ) + 2 ∞ X j =1 (cid:0) n − ( k + 1) j (cid:1) r k ( n − j ) = 0 . (6.5)Similarly, by taking Q ( q ) = ψ ( q ), we obtain nt k ( n ) + ∞ X j =1 (cid:0) n − ( k + 1) j ( j + 1) / (cid:1) t k ( n − j ( j + 1) / 2) = 0 . (6.6)We have been unable to find this result in the literature; it appears to be new.Next, we extend (6.5) by writing a recurrence for the powers of the sum side ofJacobi’s triple product identity. Example 6.4 (Powers of Jacobi’s Triple Product identity) . Let J ( q ) = 1 + ∞ X n =1 (cid:0) z n + z − n (cid:1) q n , and J n ( z ) be defined by J ( q ) r = ∞ X n =0 J r ( n, z ) q n . Applying (6.2) with s = 1 we obtain nJ r ( n, z ) + ∞ X j =1 (cid:0) n − ( r + 1) j (cid:1)(cid:0) z j + z − j (cid:1) J r ( n − j ) = 0 . (6.7)We emphasize that r need not be a positive integer in this formula.7. Application 3: Infinite families of congruences In this section we apply the results in the previous section to obtain some re-sults that are in the same vein as Ramanujan’s congruences for p (5 m + 4) and τ (5 m ) mentioned in the introduction and recent work of Chan and Wang [14]. Thefollowing theorem gives four infinite families of such congruences. Theorem 7.1. Let m be a non-negative integer and r a rational number. Let Q ( q ) = 1 / ( q ; q ) ∞ , and P r ( n ) be defined by (6.1) . Then we have the followinginfinite families of congruences.(1) P r (5 m + 1) ≡ , if r ≡ .(2) P r (5 m + 2) ≡ , if r ≡ .(3) P r (5 m + 3) ≡ , if r ≡ .(4) P r (5 m + 4) ≡ , if r ≡ . Remark. The cases r = 1 and r = − 24 in (4) give Ramanujan’s congruencesmentioned in the introduction. See Berndt [12, Ch. 2] for other proofs. A fewcongruences given by Chan and Wang [14] are also included in the above; a partof the assertions in their equations (3.1), (3.8), (3.11), (3.41), (3.43), (3.48), and,(3.49), are covered by the above. Proof. We use an inductive argument using the recurrence relation (6.4) in the form nP r ( n ) = ∞ X j =1 ( − j +1 (2 j + 1) ( n + ( r/ − j ( j + 1) / P r (cid:0) n − j ( j + 1) / (cid:1) . (7.1)First let r ≡ m = 0, (7.1) reduces to4 P r (4) = (9 + r ) P r (3) − r + 1) P r (2)so P r (4) ≡ r ≡ m > 0, consider the general term( − j +1 (2 j + 1) ( n + ( r/ − j ( j + 1) / P r (cid:0) n − j ( j + 1) / (cid:1) in (7.1) for each j . It is easy to see that when j ≡ , (cid:0) ∗ ∗ ∗ (cid:1) P r (5 m + 4 − k ) , for some k and so is ≡ j ≡ − j +1 r − P r ( ∗∗ ) (mod 5);when j ≡ − j +1 r ) P r ( ∗∗ ) (mod 5);and, when j ≡ − j +1 (9 + r ) P r ( ∗∗ ) (mod 5) . In each case, we see that when r ≡ (cid:3) By using virtually the same argument, we obtain the following congruencesmod 3. Theorem 7.2. Let m be a non-negative integer and r an integer. Let P r ( n ) bedefined by (6.1) . Then, if r ≡ , we have P r (3 m + k ) ≡ for k = 1 , . Proof. We use (7.1) again. The details are similar to the proof of Theorem 7.1. (cid:3) Concluding Remarks About his recurrence (1.1), Euler [16] (translated by Jordan Bell) wrote:Since this is the case, I seem to have advanced the science of num-bers by not a small amount when I found a certain fixed law accord-ing to which the terms of the given series 1 , , , , , etc. progress,such that by this law each term of the series can be defined from thepreceding; for I have found, which seems rather wonderful, that thisseries belongs to the kind of progression which are usually calledrecurrent and whose nature is such that each term is determined HE PARTITION-FREQUENCY ENUMERATION MATRIX 27 from the preceding according to some certain rule of relation. Andwho would have even believed that this series which is so disturbedand which seems to have nothing in common with recurrent serieswould nevertheless be included in type of series, and that it wouldbe possible to assign a scale of relation to it?Clearly, Euler’s enthusiasm for recurrence relations of this type has been shared bymany authors—including Ramanujan, Lehmer and Ewell—perhaps because theseresults provide a practical technique to compute values of the relevant functions.As we have seen, recurrences of this type can be found by considering the Partition-Frequency Enumeration matrix and its associated constructs. In addition, we haveseen some further applications which suggests that this representation is quite usefulin obtaining number-theoretic information in combinatorial contexts.Before concluding, we mention a promising direction for the future. We havenot found the moments of frequency function F k ( n ) in the theory of partitions.However, in Example 3.3, we saw that the moments M m ( n ) are related to thedivisor function σ m ( n ). Note further that the powers of the η function are relatedto the hook lengths of a partition by means of the celebrated Nekrasov-Okounkovformula (see also Han [20]); one can use this to obtain formulas for the moments, asis done by Han [19, Eq. (6.4)]. The analogues of the frequency function and theirmoments for other functions may turn out to be equally interesting. References [1] L. V. Ahlfors. Complex analysis: An introduction of the theory of analytic functions of onecomplex variable . Second edition. McGraw-Hill Book Co., New York-Toronto-London, 1966.[2] G. Andrews and M. Merca. A new generalization of Stanley’s theorem. Math. Student , 89(1-2):175–180, 2020.[3] G. E. Andrews. Number theory . W. B. Saunders Co., Philadelphia, Pa.-London-Toronto, Ont.,1971. 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