Generating Functions of Partitions
|
|
- Charlotte Johns
- 5 years ago
- Views:
Transcription
1 CHAPTER B Generating Functions of Partitions For a complex sequence {α n n 0,, 2, }, its generating function with a complex variable q is defined by A(q) : α n q n α n [q n ] A(q). When the sequence has finite non-zero terms, the generating function reduces to a polynomial. Otherwise, it becomes an infinite series. In that case, we suppose in general q < from now on. B. Basic generating functions of partitions Given three complex indeterminates x, q and n with q <, the shifted factorial is defined by (x; q) (x; q) n ( xq k ) k0 (x; q) (q n x; q). When n is a natural number in particular, it reduces to n (x; q) 0 and (x; q) n ( q k x) for n, 2,. k0 We shall frequently use the following abbreviated notation for shifted factorial fraction: [ ] a, b,, c q (a; q) n(b; q) n (c; q) n. α, β,, γ n (α; q) n (β; q) n (γ; q) n
2 2 CHU Wenchang and DI CLAUDIO Leontina B.. Partitions with parts in S. We first investigate the generating functions of partitions with parts in S, where the basic set S N with N being the set of natural numbers. Let S be a set of natural numbers and p(n S) denote the number of partitions of n into elements of S (or in other words, the parts of partitions belong to S). Then the univariate generating function is given by p(n S) q n q k. (B.a) If we denote further by p l (n S) the number of partitions with exactly l-parts in S, then the bivariate generating function is p l (n S) x l q n xq. (B.b) k l,n 0 Proof. For q <, we can expand the right member of the equation (B.a) according to the geometric series q kmk q kmk. q k m k0 m k 0 Extracting the coefficient of q n from both sides, we obtain [q n ] q k [qn ] q kmk m k 0 kmkn m k 0: The last sum is equal to the number of solutions of the Diophantine equation km k n which enumerates the partitions { m, 2 m2,,n mn } of n into parts in S.. This completes the proof of (B.a). (B.b) can be verified similarly. The bivariate generating function In fact, consider the formal power series expansion x mk q kmk x xq k mk q kmk m k0 m k 0
3 Classical Partition Identities and Basic Hypergeometric Series 3 in which the coefficient of x l q n reads as [x l q n ] xq k [x l q n ] x mk q kmk m k 0. m } k l km k n m k 0: The last sum enumerates the solutions of the system of Diophantine equations m k l km k n which are the number of partitions { } m, 2 m2,,n mn of n with exactly l-parts in S. B.2. Partitions into distinct parts in S. Next we study the generating functions of partitions into distinct parts in S. If we denote by Q(n S) and Q l (n S) the corresponding partition numbers with distinct parts from S, then their generating functions read respectively as Q(n S) q n ( +q k ) (B.2a) ( ) +xq k. (B.2b) l,n 0 Q l (n S) x l q n Proof. For the first identity, observing that +q k m k0, q kmk we can reformulate the product on the right hand side as ( ) +q k q kmk q kmk. m k0, m k0, Extracting the coefficient of q n, we obtain [q n ] ( ) +q k [q n ] q kmk m k0, kmkn m k0,:.
4 4 CHU Wenchang and DI CLAUDIO Leontina The last sum enumerates the solutions of Diophantine equation km k n with m k 0, which is equal to Q(n S), the number of partitions of n into distinct parts in S. Instead, we can proceed similarly for the second formula as follows: ( + xq k ) m k0, x mk q kmk m k0, x mk q kmk. The coefficient of x l q n leads us to the following [x l q n ] ( + xq k ) [x l q n ] m k0, m k l km k n m k0,: x mk q kmk. } The last sum equals the number of solutions of the system of Diophantine equations m k l km k n with m k 0, which correspond to the partitions { } m, 2 m2,,n mn of n with exactly l distinct parts in S.
5 Classical Partition Identities and Basic Hypergeometric Series 5 B.3. Classical generating functions. When S N, the set of natural numbers, the corresponding generating functions may be displayed, respectively, as (q; q) (qx; q) ( q; q) ( qx; q) m m q m p(n) q n xq m p l (n) x l q n ( + q m ) m m l,n 0 Q(n) q n ( + xq m ) Q l (n) x l q n. l,n 0 (B.3a) (B.3b) (B.3c) (B.3d) Manipulating the generating function of the partitions into odd numbers in the following manner k q 2k k k q k q 2k q k k ( q 2k ) ( + q k ) k we see that it results in the generating function of the partitions into distinct parts. We have therefore proved the following theorem due to Euler. The number of partitions of n into odd numbers equals to the number of partitions of n into distinct parts. B2. Classical partitions and the Gauss formula B2.. Proposition. Let p m (n) be the number of partitions into exactly m parts (or dually, partitions with the largest part equal to m). Its generating function reads as p m (n) q n q m ( q)( q 2 ) ( q m ). (B2.)
6 6 CHU Wenchang and DI CLAUDIO Leontina Proof. For S N, the generating function of {p l (n N)} reads as l (n) x l,n 0p l q n x l p l (n)q n l 0 n 0 xq k. (qx; q) k Extracting the coefficient of x m, we get p m (n) q n [x m ]. (qx; q) For q <, the function /(qx; q) is analytic at x 0. We can therefore expand it in MacLaurin series: A l (q)x l (B2.2) (qx; q) where the coefficients { A l (q) } are independent of x to be determined. Performing the replacement x x/q, we can restate the expansion just displayed as A l (q)x l q l. (x; q) l0 (B2.3) It is evident that (B2.2) equals ( x) times (B2.3), which results in the functional equation A l (q)x l ( x) A l (q)x l q l. l0 Extracting the coefficient of x m from both expansions, we get A m (q) A m (q)q m A m (q)q m which is equivalent to the following recurrence relation q A m (q) q m A m (q) where m, 2,. Iterating this recursion for m-times, we find that q m A 0 (q) A m (q) ( q m )( q m ) ( q) qm A 0 (q). (q; q) m Noting that A 0 (q), we get finally p m (n) q n [x m ] qm. (qx; q) (q; q) m This completes the proof of Proposition B2.. l0 l0
7 Classical Partition Identities and Basic Hypergeometric Series 7 A combinatorial proof. Let p(n λ m) be the number of partitions of n with the first part λ equal to m. Then p m (n) p(n λ m) because the partitions enumerated by p m (n) are conjugate with those enumerated by p(n λ m). Therefore they have the same generating functions: p m (n) q n p(n λ m) q n. All the partition of n enumerated by p(n λ m) have the first part λ m in common and the remaining parts constitute the partitions of n m with each part m. Therefore we have p(n λ m) q n p(n m λ m) q n q m p(n λ m) q n nm q m p(n {, 2,,m}) q n qm (q; q) m where the first line is justified by replacement n n + m on summation index, while the second is a consequence of (B.a). This confirms again the generating function (B2.). B2.2. Proposition. Let p m (n) be the number of partitions into m parts (or dually, partitions into parts m). Then we have the generating function p m (n) q n which yields a finite summation formula ( q)( q 2 ) ( q m ) m ( q)( q 2 ) ( q m ) + q k ( q)( q 2 ) ( q k ). k (B2.4) Proof. Notice that p m (n), the number of partitions into m parts is equal to the number of partitions into parts m in view of conjugate partitions. We get immediately from (B.a) the generating function (B2.4). The classification of the partitions of n into m parts with respect to the number k of parts yields p m (n) p 0 (n)+p (n)+p 2 (n)+ + p m (n).
8 8 CHU Wenchang and DI CLAUDIO Leontina The corresponding generating function results in p m (n) q n m p k (n) q n k0 m k0 q k (q; q) k. Recalling the first generating function expression (B2.4), we get the second formula from the last relation. B2.3. Gauss classical partition identity. xq n + m x m ( q)( q 2 ) ( q m ). (B2.5) Proof. In fact, we have already established this identity from the demonstration of the last theorem, where it has been displayed explicitly in (B2.3). Alternatively, classifying all the partitions with respect to the number of parts, we can manipulate the bivariate generating function (xq; q) l, l0 p l (n)x l q n x l l0 x l q l ( q)( q 2 ) ( q l ) which is equivalent to Gauss classical partition identity. p l (n)q n B2.4. Theorem. Let p l (n m) be the number of partitions of n with exactly l-parts m. Then we have its generating function l, p l (n m) x l q n ( qx)( q 2 x) ( q m x). The classification with respect to the maximum part k of partitions produces another identity m ( qx)( q 2 x) ( q m x) +x q k ( qx)( q 2 x) ( q k x). k Proof. The first generating function follows from (B.b).
9 Classical Partition Identities and Basic Hypergeometric Series 9 From the first generating function, we see that the bivariate generating function of partitions into parts k reads as l, p l (n k) x l q n (qx; q) k. Putting an extra part λ k with enumerator xq k over the partitions enumerated by the last generating function, we therefore derive the bivariate generating function of partitions into l parts with the first one λ k as follows: p l (n λ k) x l q n xqk. (qx; q) k l, Classifying the partitions of n into exactly l parts with each parts m according to the first part λ k, we get the following expression l, p l (n m) x l q n m k0 l, +x m k p l (n λ k)x l q n q k (qx; q) k which is the second identity. B3. Partitions into distinct parts and the Euler formula B3.. Theorem. Let Q m (n) be the number of partitions into exactly m distinct parts. Its generating function reads as Q m (n) q n q (+m 2 ) ( q)( q 2 ) ( q m ). (B3.) Proof. Let λ (λ >λ 2 >λ m > 0) be a partition enumerated by Q m (n). Based on λ, define another partition µ (µ µ 2 µ m 0) by µ k : λ k (m k + ) for k, 2,,m. (B3.2)
10 20 CHU Wenchang and DI CLAUDIO Leontina It is obvious that µ is a partition of λ ( +m 2 ) into m parts. As an example, the following figures show this correspondence between two partitions λ (9743) and µ (43). λ (9743) µ (43) It is not difficult to verify that the mapping (B3.2) is a bijection between the partitions of n with exactly m distinct parts and the partitions of n ( +m 2 ) with m parts. Therefore the generating function of {Q m (n)} n is equal to that of {p m( n ( +m 2 ) ) } n, the number of partitions of n ( +m 2 ) with the number of parts m: Q m (n) q n q (+m 2 ) p m (n) q n p m( n ( +m 2 ) ) q n q(+m 2 ) (q; q) m thanks for the generating function displayed in (B2.4). This completes the proof of Theorem B3.. Instead of the ordinary Ferrers diagram, we can draw a shifted diagram of λ as follows (see the figure). Under the first row of λ squares, we put λ 2 squares lined up vertically from the second column. For the third row, we
11 Classical Partition Identities and Basic Hypergeometric Series 2 put λ 3 squares beginning from the third column. Continuing in this way, the last row of λ m squares will be lined up vertically from the m-th column. The shifted diagram of partition λ (9743) From the shifted diagram of λ, we see that all the partitions enumerated by Q m (n) have one common triangle on the left whose weight is ( +m 2 ). The remaining parts right to the triangle are partitions of n ( m+ 2 ) with m parts. This reduces the problem of computing the generating function to the case just explained. B3.2. Classifying all the partitions with distinct parts according to the number of parts, we get Euler s classical partition identity ( xq n ) + m ( ) m x m q (m 2 ) ( q)( q 2 ) ( q m ) (B3.3) which can also be verified through the correspondence between partitions into distinct odd parts and self-conjugate partitions. Proof. Considering the bivariate generating function of Q m (n), we have ( + xq k ) k m0 x m Q m (n)q n.
12 22 CHU Wenchang and DI CLAUDIO Leontina Recalling (B3.) and then noting that Q 0 (n) δ 0,n, we deduce that ( + xq k ) + k m x m q (+m 2 ) (q; q) m which becomes the Euler identity under parameter replacement x x/q. In view of Euler s Theorem A2., we have a bijection between the partitions into distinct odd parts and the self-conjugate partitions. The self-conjugate partition λ (65322) with the Durfee square 3 3 For a self-conjugate partition with the main diagonal length equal to m (which corresponds exactly to the length of partitions into distinct odd parts), it consists of three pieces: the first piece is the square of m m on the top-left with bivariate enumerator x m q m2, the second piece right to the square is a partition with m parts enumerated by /(q; q) m and the third piece under the square is in effect the conjugate of the second one. Therefore the partitions right to the square and under the square m m are altogether enumerated by /(q 2 ; q 2 ) m. Classifying the self-conjugate partitions according to the main diagonal length m, multiplying both generating functions together and summing m
13 Classical Partition Identities and Basic Hypergeometric Series 23 over 0 m, we find the following identity: ( + xq +2n ) m0 x m q m2 (q 2 ; q 2 ) m where the left hand side is the bivariate generating function of the partitions into odd distinct parts. It is trivial to verify that under replacements x xq /2 and q q /2 the last formula is exactly the identity displayed in (B3.3). Unfortunately, there does not exist the closed form for the generating function of Q m (n), numbers of partitions into m distinct parts. B3.3. Dually, if we classify the partitions into distinct parts m according to their maximum part. Then we can derive the following finite and infinite series identities m m k ( + q j x) +x q k ( + q i x) (B3.4a) j j k ( + q j x) +x k i ( + q i x). (B3.4b) q k k i Proof. For the partitions into distinct parts with the maximum part equal to k, their bivariate generating function is given by k q k x ( + q i x) which reduces to for k 0. i Classifying the partitions into distinct parts m according to their maximum part k with 0 k m, weget m ( qx; q) m +x q k ( qx; q) k. k The second identity follows from the first one with m.
14 24 CHU Wenchang and DI CLAUDIO Leontina B4. Partitions and the Gauss q-binomial coefficients B4.. Lemma. Let p l (n m) and p l (n m) be the numbers of partitions of n into l and l parts, respectively, with each part m. We have the generating functions: l,n 0 l,n 0 p l (n m) x l q n p l (n m) x l q n ( xq)( xq 2 ) ( xq m ) (B4.a) ( x)( xq) ( xq m ). (B4.b) The first identity (B4.a) is a special case of the generating function shown in (B.b). On account of the length of partitions, we have p l (n m) p 0 (n m)+p (n m)+ + p l (n m). Manipulating the triple sum and then applying the geometric series, we can calculate the corresponding generating function as follows: l,n 0 p l (n m) x l q n l,n 0 k0 k0 x l p k (n m)x l q n p k (n m)q n k0 x l lk p k (n m)x k q n. The last expression leads us immediately to the second bivariate generating function (B4.b) in view of the first generating function (B4.a). B4.2. The Gauss q-binomial coefficients as generating functions. Let p l (n m) and p l (n m) be as in Lemma B4.. The corresponding univariate generating functions read respectively as [ ] l + m p l (n m) q n q l (B4.2a) m n 0 [ ] l + m p l (n m) q n (B4.2b) m n 0
15 Classical Partition Identities and Basic Hypergeometric Series 25 where the q-gauss binomial coefficient is defined by [ ] m + n (q; q) m+n. m (q; q) m (q; q) n q Proof. For these two formulae, it is sufficient to prove only one identity because p l (n m) p l (n m) p l (n m). In fact, supposing that (B4.2b) is true, then (B4.2a) follows in this manner: p l (n m) q n p l (n m) q n p l (n m) q n n 0 n 0 n 0 [ ] [ ] l + m l +m m m q q q l [ l + m m ]. q Now we should prove (B4.2b). Extracting the coefficient of x l from the generation function (B4.b), we get p l (n m) q n [x l ]. (x; q) m+ Observing that the function /(x; q) m+ is analytic at x 0 for q <, we can expand it into MacLaurin series: B k (q)x k (x; q) m+ where the coefficients { B k (q) } are independent of x to be determinated. Reformulating it under replacement x qx as B k (q)x k q k (qx; q) m+ and then noting further that both fractions just displayed differ in factors ( x) and ( xq m+ ), we have accordingly the following: ( x) B k (q)x k ( xq m+ ) B k (q)x k q k. k0 k0 k0 k0 Extracting the coefficient of x l from both sides we get B l (q) B l (q) q l B l (q) q m+l B l (q) which is equivalent to the following recurrence relation B l (q) B l (q) qm+l q l for l, 2,.
16 26 CHU Wenchang and DI CLAUDIO Leontina Iterating this relation l-times, we find that B l (q) B 0 (q) (qm+ ; q) l (q; q) l [ ] m + l l q where B 0 (q) follows from setting x 0 in the generating function (xq; q) m+ B k (q)x k. k0 Therefore we conclude the proof. B4.3. Theorem. Classifying the partitions according to the number of parts, we derive immediately two q-binomial identities (finite and infinite): n [ ] l + m q l m l0 [ ] l + m x l m l0 [ ] m + n + n (B4.3a) m xq k. (B4.3b) k0 Proof. In view of (B4.2a) and (B4.2b), the univariate generating functions for the partitions into parts m + with the lengths equal to l and [ ] l + m n are respectively given by the q-binomial coefficients q l and m [ ] m + n +. Classifying the partitions enumerated by the latter according to the number of parts l with 0 l n, we establish the first n identity. By means of (B4.b), we have m xq k k0 x l l0 p l (n m) q n l0 [ ] m + l x l l which is the second q-binomial identity.
17 Classical Partition Identities and Basic Hypergeometric Series 27 B5. Partitions into distinct parts and finite q-differences Similarly, let Q l (n m) be the number of partitions of n into exactly l distinct parts with each part m. Then we have generating functions [ ] m q (+l 2 ) Q l (n m) q n (B5.) l n 0 m ( + xq k ) Q l (n m) x l q n k l,n 0 (B5.2) whose combination leads us to Euler s finite q-differences n (x; q) n ( xq l ) l0 k0 n [ ] n ( ) k q (k 2) x k. k (B5.3) Following the second proof of Theorem B3., we can check without difficulty that the shifted Ferrers diagrams of the partitions into l-parts m are unions of the same triangle of length l enumerated by q (l+ 2 ) and the ordinary partitions into parts m l with length l whose generating function [ m ] reads as the q-binomial coefficient. The product of them gives the l generating function for {Q l (n m)} n. The second formula is a particular case of (B.2b). Its combination with the univariate generating function just proved leads us to the following: m m [ ] m ( qx; q) m Q l (n m) q n x l x l q (+l 2 ). l l0 n 0 Replacing x by x/q in the above, we get Euler s q-difference formula: m [ ] m (x; q) m ( ) l q (l 2) x l. l l0 l0 Remark The last formula is called the Euler q-difference formula because if we put x : q n, the finite sum results in m [ ] { m 0, 0 n<m ( ) l q (l 2) ln (q n ; q) m l ( ) n q (n+ 2 ) (q; q)n, n m l0 just like the ordinary finite differences of polynomials.
18 28 CHU Wenchang and DI CLAUDIO Leontina Keep in mind of the q-binomial limit [ ] n (q+n k ; q) k as n. k (q; q) k (q; q) k Letting n in Euler s q-finite differences, we recover again the Euler classical partition identity ( ) k x k (x; q) q 2) (k (q; q) k k0 where Tannery s theorem has been applied for the limiting process.
Congruence Properties of Partition Function
CHAPTER H Congruence Properties of Partition Function Congruence properties of p(n), the number of partitions of n, were first discovered by Ramanujan on examining the table of the first 200 values of
More informationCLASSICAL PARTITION IDENTITIES AND BASIC HYPERGEOMETRIC SERIES
UNIVERSITÀ DEGLI STUDI DI LECCE DIPARTIMENTO DI MATEMATICA "Ennio De Giorgi" Wenchang Chu Leontina Di Claudio Dipartimento di Matematica Università di Lecce Lecce - Italia CLASSICAL PARTITION IDENTITIES
More informationPartitions and Algebraic Structures
CHAPTER A Partitions and Algebraic Structures In this chapter, we introduce partitions of natural numbers and the Ferrers diagrams. The algebraic structures of partitions such as addition, multiplication
More informationApplicable Analysis and Discrete Mathematics available online at ABEL S METHOD ON SUMMATION BY PARTS.
Applicable Analysis and Discrete Mathematics available online at http://pefmathetfrs Appl Anal Discrete Math 4 010), 54 65 doi:1098/aadm1000006c ABEL S METHOD ON SUMMATION BY PARTS AND BALANCED -SERIES
More informationInteger Partitions With Even Parts Below Odd Parts and the Mock Theta Functions
Integer Partitions With Even Parts Below Odd Parts and the Mock Theta Functions by George E. Andrews Key Words: Partitions, mock theta functions, crank AMS Classification Numbers: P84, P83, P8, 33D5 Abstract
More informationMULTISECTION METHOD AND FURTHER FORMULAE FOR π. De-Yin Zheng
Indian J. pure appl. Math., 39(: 37-55, April 008 c Printed in India. MULTISECTION METHOD AND FURTHER FORMULAE FOR π De-Yin Zheng Department of Mathematics, Hangzhou Normal University, Hangzhou 30036,
More informationGenerating Functions
Semester 1, 2004 Generating functions Another means of organising enumeration. Two examples we have seen already. Example 1. Binomial coefficients. Let X = {1, 2,..., n} c k = # k-element subsets of X
More informationThesis submitted in partial fulfillment of the requirement for The award of the degree of. Masters of Science in Mathematics and Computing
SOME n-color COMPOSITION Thesis submitted in partial fulfillment of the requirement for The award of the degree of Masters of Science in Mathematics and Computing Submitted by Shelja Ratta Roll no- 301203014
More informationChapter 1. Sets and Numbers
Chapter 1. Sets and Numbers 1. Sets A set is considered to be a collection of objects (elements). If A is a set and x is an element of the set A, we say x is a member of A or x belongs to A, and we write
More informationWhat you learned in Math 28. Rosa C. Orellana
What you learned in Math 28 Rosa C. Orellana Chapter 1 - Basic Counting Techniques Sum Principle If we have a partition of a finite set S, then the size of S is the sum of the sizes of the blocks of the
More informationPartitions, permutations and posets Péter Csikvári
Partitions, permutations and posets Péter Csivári In this note I only collect those things which are not discussed in R Stanley s Algebraic Combinatorics boo Partitions For the definition of (number) partition,
More informationq-series Michael Gri for the partition function, he developed the basic idea of the q-exponential. From
q-series Michael Gri th History and q-integers The idea of q-series has existed since at least Euler. In constructing the generating function for the partition function, he developed the basic idea of
More informationDifference Equation on Quintuple Products and Ramanujan s Partition Congruence p(11n +6) 0 (mod 11)
International Mathematical Forum, 5, 2010, no 31, 1533-1539 Difference Equation on Quintuple Products and Ramanujan s Partition Congruence p(11n +6) 0 (mod 11) Qinglun Yan 1, Xiaona Fan and Jing Fu College
More informationHypergeometric series and the Riemann zeta function
ACTA ARITHMETICA LXXXII.2 (997) Hypergeometric series and the Riemann zeta function by Wenchang Chu (Roma) For infinite series related to the Riemann zeta function, De Doelder [4] established numerous
More informationTwo contiguous relations of Carlitz and Willett for balanced series Wenchang Chu and Xiaoyuan Wang
Lecture Notes of Seminario Interdisciplinare di Matematica Vol 9(200), pp 25 32 Two contiguous relations of Carlitz and Willett for balanced series Wenchang Chu and Xiaoyuan Wang Abstract The modified
More informationare the q-versions of n, n! and . The falling factorial is (x) k = x(x 1)(x 2)... (x k + 1).
Lecture A jacques@ucsd.edu Notation: N, R, Z, F, C naturals, reals, integers, a field, complex numbers. p(n), S n,, b(n), s n, partition numbers, Stirling of the second ind, Bell numbers, Stirling of the
More informationAlgorithmic Approach to Counting of Certain Types m-ary Partitions
Algorithmic Approach to Counting of Certain Types m-ary Partitions Valentin P. Bakoev Abstract Partitions of integers of the type m n as a sum of powers of m (the so called m-ary partitions) and their
More informationNOTES ON DIOPHANTINE APPROXIMATION
NOTES ON DIOPHANTINE APPROXIMATION Jan-Hendrik Evertse January 29, 200 9 p-adic Numbers Literature: N. Koblitz, p-adic Numbers, p-adic Analysis, and Zeta-Functions, 2nd edition, Graduate Texts in Mathematics
More informationChapter Generating Functions
Chapter 8.1.1-8.1.2. Generating Functions Prof. Tesler Math 184A Fall 2017 Prof. Tesler Ch. 8. Generating Functions Math 184A / Fall 2017 1 / 63 Ordinary Generating Functions (OGF) Let a n (n = 0, 1,...)
More information= i 0. a i q i. (1 aq i ).
SIEVED PARTITIO FUCTIOS AD Q-BIOMIAL COEFFICIETS Fran Garvan* and Dennis Stanton** Abstract. The q-binomial coefficient is a polynomial in q. Given an integer t and a residue class r modulo t, a sieved
More informationSome Results Concerning Uniqueness of Triangle Sequences
Some Results Concerning Uniqueness of Triangle Sequences T. Cheslack-Postava A. Diesl M. Lepinski A. Schuyler August 12 1999 Abstract In this paper we will begin by reviewing the triangle iteration. We
More informationThe Bhargava-Adiga Summation and Partitions
The Bhargava-Adiga Summation and Partitions By George E. Andrews September 12, 2016 Abstract The Bhargava-Adiga summation rivals the 1 ψ 1 summation of Ramanujan in elegance. This paper is devoted to two
More informationA matrix over a field F is a rectangular array of elements from F. The symbol
Chapter MATRICES Matrix arithmetic A matrix over a field F is a rectangular array of elements from F The symbol M m n (F ) denotes the collection of all m n matrices over F Matrices will usually be denoted
More informationELEMENTARY LINEAR ALGEBRA
ELEMENTARY LINEAR ALGEBRA K R MATTHEWS DEPARTMENT OF MATHEMATICS UNIVERSITY OF QUEENSLAND First Printing, 99 Chapter LINEAR EQUATIONS Introduction to linear equations A linear equation in n unknowns x,
More informationAn Involution for the Gauss Identity
An Involution for the Gauss Identity William Y. C. Chen Center for Combinatorics Nankai University, Tianjin 300071, P. R. China Email: chenstation@yahoo.com Qing-Hu Hou Center for Combinatorics Nankai
More informationq GAUSS SUMMATION VIA RAMANUJAN AND COMBINATORICS
q GAUSS SUMMATION VIA RAMANUJAN AND COMBINATORICS BRUCE C. BERNDT 1 and AE JA YEE 1. Introduction Recall that the q-gauss summation theorem is given by (a; q) n (b; q) ( n c ) n (c/a; q) (c/b; q) =, (1.1)
More informationPURE MATHEMATICS AM 27
AM SYLLABUS (2020) PURE MATHEMATICS AM 27 SYLLABUS 1 Pure Mathematics AM 27 (Available in September ) Syllabus Paper I(3hrs)+Paper II(3hrs) 1. AIMS To prepare students for further studies in Mathematics
More informationGenerating Functions (Revised Edition)
Math 700 Fall 06 Notes Generating Functions (Revised Edition What is a generating function? An ordinary generating function for a sequence (a n n 0 is the power series A(x = a nx n. The exponential generating
More informationOn q-series Identities Arising from Lecture Hall Partitions
On q-series Identities Arising from Lecture Hall Partitions George E. Andrews 1 Mathematics Department, The Pennsylvania State University, University Par, PA 16802, USA andrews@math.psu.edu Sylvie Corteel
More informationThe 4-periodic spiral determinant
The 4-periodic spiral determinant Darij Grinberg rough draft, October 3, 2018 Contents 001 Acknowledgments 1 1 The determinant 1 2 The proof 4 *** The purpose of this note is to generalize the determinant
More informationPartition Numbers. Kevin Y.X. Wang. September 5, School of Mathematics and Statistics The University of Sydney
Partition Numbers Kevin Y.X. Wang School of Mathematics and Statistics The University of Sydney September 5, 2017 Outline 1 Introduction 2 Generating function of partition numbers 3 Jacobi s Triple Product
More informationCHAPTER 1 POLYNOMIALS
1 CHAPTER 1 POLYNOMIALS 1.1 Removing Nested Symbols of Grouping Simplify. 1. 4x + 3( x ) + 4( x + 1). ( ) 3x + 4 5 x 3 + x 3. 3 5( y 4) + 6 y ( y + 3) 4. 3 n ( n + 5) 4 ( n + 8) 5. ( x + 5) x + 3( x 6)
More informationand the compositional inverse when it exists is A.
Lecture B jacques@ucsd.edu Notation: R denotes a ring, N denotes the set of sequences of natural numbers with finite support, is a generic element of N, is the infinite zero sequence, n 0 R[[ X]] denotes
More informationSolving Differential Equations Using Power Series
LECTURE 25 Solving Differential Equations Using Power Series We are now going to employ power series to find solutions to differential equations of the form (25.) y + p(x)y + q(x)y = 0 where the functions
More informationSolving Differential Equations Using Power Series
LECTURE 8 Solving Differential Equations Using Power Series We are now going to employ power series to find solutions to differential equations of the form () y + p(x)y + q(x)y = 0 where the functions
More informationTILING PROOFS OF SOME FORMULAS FOR THE PELL NUMBERS OF ODD INDEX
#A05 INTEGERS 9 (2009), 53-64 TILING PROOFS OF SOME FORMULAS FOR THE PELL NUMBERS OF ODD INDEX Mark Shattuck Department of Mathematics, University of Tennessee, Knoxville, TN 37996-1300 shattuck@math.utk.edu
More informationa 11 x 1 + a 12 x a 1n x n = b 1 a 21 x 1 + a 22 x a 2n x n = b 2.
Chapter 1 LINEAR EQUATIONS 11 Introduction to linear equations A linear equation in n unknowns x 1, x,, x n is an equation of the form a 1 x 1 + a x + + a n x n = b, where a 1, a,, a n, b are given real
More informationStirling Numbers of the 1st Kind
Daniel Reiss, Colebrook Jackson, Brad Dallas Western Washington University November 28, 2012 Introduction The set of all permutations of a set N is denoted S(N), while the set of all permutations of {1,
More informationGeneralized Akiyama-Tanigawa Algorithm for Hypersums of Powers of Integers
1 2 3 47 6 23 11 Journal of Integer Sequences, Vol 16 (2013, Article 1332 Generalized Aiyama-Tanigawa Algorithm for Hypersums of Powers of Integers José Luis Cereceda Distrito Telefónica, Edificio Este
More informationUNIFICATION OF THE QUINTUPLE AND SEPTUPLE PRODUCT IDENTITIES. 1. Introduction and Notation
UNIFICATION OF THE QUINTUPLE AND SEPTUPLE PRODUCT IDENTITIES WENCHANG CHU AND QINGLUN YAN Department of Applied Mathematics Dalian University of Technology Dalian 116024, P. R. China Abstract. By combining
More informationThe Perrin Conjugate and the Laguerre Orthogonal Polynomial
The Perrin Conjugate and the Laguerre Orthogonal Polynomial In a previous chapter I defined the conjugate of a cubic polynomial G(x) = x 3 - Bx Cx - D as G(x)c = x 3 + Bx Cx + D. By multiplying the polynomial
More informationAN ELEMENTARY APPROACH TO THE MACDONALD IDENTITIES* DENNIS STANTON
AN ELEMENTARY APPROACH TO THE MACDONALD IDENTITIES* DENNIS STANTON Abstract. Elementary proofs are given for the infinite families of Macdonald identities. The reflections of the Weyl group provide sign-reversing
More informationParity Results for Certain Partition Functions
THE RAMANUJAN JOURNAL, 4, 129 135, 2000 c 2000 Kluwer Academic Publishers. Manufactured in The Netherlands. Parity Results for Certain Partition Functions MICHAEL D. HIRSCHHORN School of Mathematics, UNSW,
More information#A5 INTEGERS 18A (2018) EXPLICIT EXAMPLES OF p-adic NUMBERS WITH PRESCRIBED IRRATIONALITY EXPONENT
#A5 INTEGERS 8A (208) EXPLICIT EXAMPLES OF p-adic NUMBERS WITH PRESCRIBED IRRATIONALITY EXPONENT Yann Bugeaud IRMA, UMR 750, Université de Strasbourg et CNRS, Strasbourg, France bugeaud@math.unistra.fr
More informationSection Taylor and Maclaurin Series
Section.0 Taylor and Maclaurin Series Ruipeng Shen Feb 5 Taylor and Maclaurin Series Main Goal: How to find a power series representation for a smooth function us assume that a smooth function has a power
More information(Inv) Computing Invariant Factors Math 683L (Summer 2003)
(Inv) Computing Invariant Factors Math 683L (Summer 23) We have two big results (stated in (Can2) and (Can3)) concerning the behaviour of a single linear transformation T of a vector space V In particular,
More information5.4 Bessel s Equation. Bessel Functions
SEC 54 Bessel s Equation Bessel Functions J (x) 87 # with y dy>dt, etc, constant A, B, C, D, K, and t 5 HYPERGEOMETRIC ODE At B (t t )(t t ), t t, can be reduced to the hypergeometric equation with independent
More informationMore about partitions
Partitions 2.4, 3.4, 4.4 02 More about partitions 3 + +, + 3 +, and + + 3 are all the same partition, so we will write the numbers in non-increasing order. We use greek letters to denote partitions, often
More informationarxiv: v2 [math.nt] 9 Apr 2015
CONGRUENCES FOR PARTITION PAIRS WITH CONDITIONS arxiv:408506v2 mathnt 9 Apr 205 CHRIS JENNINGS-SHAFFER Abstract We prove congruences for the number of partition pairs π,π 2 such that π is nonempty, sπ
More informationn] (q; q)n (2) k - (q; q)k (q; q)n-k.
Convegno Nazionale Matematica senza Frontiere Lecce, 5-8 marzo 2003 Espansioni Binomiali Non-Commutative e Relazioni di Serie Inverse CRU Wenchang, ZRANG Zhizheng Dipartimento di Matematica "E. De Giorgi"-
More informationCHAPTER 4: EXPLORING Z
CHAPTER 4: EXPLORING Z MATH 378, CSUSM. SPRING 2009. AITKEN 1. Introduction In this chapter we continue the study of the ring Z. We begin with absolute values. The absolute value function Z N is the identity
More informationNotes on Continued Fractions for Math 4400
. Continued fractions. Notes on Continued Fractions for Math 4400 The continued fraction expansion converts a positive real number α into a sequence of natural numbers. Conversely, a sequence of natural
More informationWeek 9-10: Recurrence Relations and Generating Functions
Week 9-10: Recurrence Relations and Generating Functions April 3, 2017 1 Some number sequences An infinite sequence (or just a sequence for short is an ordered array a 0, a 1, a 2,..., a n,... of countably
More informationReview of Power Series
Review of Power Series MATH 365 Ordinary Differential Equations J. Robert Buchanan Department of Mathematics Fall 2018 Introduction In addition to the techniques we have studied so far, we may use power
More informationPURE MATHEMATICS AM 27
AM Syllabus (014): Pure Mathematics AM SYLLABUS (014) PURE MATHEMATICS AM 7 SYLLABUS 1 AM Syllabus (014): Pure Mathematics Pure Mathematics AM 7 Syllabus (Available in September) Paper I(3hrs)+Paper II(3hrs)
More information2 Lecture 2: Logical statements and proof by contradiction Lecture 10: More on Permutations, Group Homomorphisms 31
Contents 1 Lecture 1: Introduction 2 2 Lecture 2: Logical statements and proof by contradiction 7 3 Lecture 3: Induction and Well-Ordering Principle 11 4 Lecture 4: Definition of a Group and examples 15
More informationCounting k-marked Durfee Symbols
Counting k-marked Durfee Symbols Kağan Kurşungöz Department of Mathematics The Pennsylvania State University University Park PA 602 kursun@math.psu.edu Submitted: May 7 200; Accepted: Feb 5 20; Published:
More informationLinear Algebra. Linear Equations and Matrices. Copyright 2005, W.R. Winfrey
Copyright 2005, W.R. Winfrey Topics Preliminaries Systems of Linear Equations Matrices Algebraic Properties of Matrix Operations Special Types of Matrices and Partitioned Matrices Matrix Transformations
More informationarxiv: v1 [math.co] 21 Sep 2015
Chocolate Numbers arxiv:1509.06093v1 [math.co] 21 Sep 2015 Caleb Ji, Tanya Khovanova, Robin Park, Angela Song September 22, 2015 Abstract In this paper, we consider a game played on a rectangular m n gridded
More informationIntroduction to Number Theory
INTRODUCTION Definition: Natural Numbers, Integers Natural numbers: N={0,1,, }. Integers: Z={0,±1,±, }. Definition: Divisor If a Z can be writeen as a=bc where b, c Z, then we say a is divisible by b or,
More informationOn Partition Functions and Divisor Sums
1 2 3 47 6 23 11 Journal of Integer Sequences, Vol. 5 (2002), Article 02.1.4 On Partition Functions and Divisor Sums Neville Robbins Mathematics Department San Francisco State University San Francisco,
More informationMULTI-RESTRAINED STIRLING NUMBERS
MULTI-RESTRAINED STIRLING NUMBERS JI YOUNG CHOI DEPARTMENT OF MATHEMATICS SHIPPENSBURG UNIVERSITY SHIPPENSBURG, PA 17257, U.S.A. Abstract. Given positive integers n, k, and m, the (n, k)-th m- restrained
More informationSeries Solutions. 8.1 Taylor Polynomials
8 Series Solutions 8.1 Taylor Polynomials Polynomial functions, as we have seen, are well behaved. They are continuous everywhere, and have continuous derivatives of all orders everywhere. It also turns
More informationLEGENDRE S THEOREM, LEGRANGE S DESCENT
LEGENDRE S THEOREM, LEGRANGE S DESCENT SUPPLEMENT FOR MATH 370: NUMBER THEORY Abstract. Legendre gave simple necessary and sufficient conditions for the solvablility of the diophantine equation ax 2 +
More informationMATH 117 LECTURE NOTES
MATH 117 LECTURE NOTES XIN ZHOU Abstract. This is the set of lecture notes for Math 117 during Fall quarter of 2017 at UC Santa Barbara. The lectures follow closely the textbook [1]. Contents 1. The set
More informationArticle 12 INTEGERS 11A (2011) Proceedings of Integers Conference 2009 RECURSIVELY SELF-CONJUGATE PARTITIONS
Article 12 INTEGERS 11A (2011) Proceedings of Integers Conference 2009 RECURSIVELY SELF-CONJUGATE PARTITIONS William J. Keith Department of Mathematics, Drexel University, Philadelphia, PA 19104, USA wjk26@math.drexel.edu
More informationBasis Partitions. Herbert S. Wilf Department of Mathematics University of Pennsylvania Philadelphia, PA Abstract
Jennifer M. Nolan Department of Computer Science North Carolina State University Raleigh, North Carolina 27695-8206 Basis Partitions Herbert S. Wilf Department of Mathematics University of Pennsylvania
More informationQUADRANT MARKED MESH PATTERNS IN 132-AVOIDING PERMUTATIONS III
#A39 INTEGERS 5 (05) QUADRANT MARKED MESH PATTERNS IN 3-AVOIDING PERMUTATIONS III Sergey Kitaev Department of Computer and Information Sciences, University of Strathclyde, Glasgow, United Kingdom sergey.kitaev@cis.strath.ac.uk
More informationq-counting hypercubes in Lucas cubes
Turkish Journal of Mathematics http:// journals. tubitak. gov. tr/ math/ Research Article Turk J Math (2018) 42: 190 203 c TÜBİTAK doi:10.3906/mat-1605-2 q-counting hypercubes in Lucas cubes Elif SAYGI
More informationSome families of identities for the integer partition function
MATHEMATICAL COMMUNICATIONS 193 Math. Commun. 0(015), 193 00 Some families of identities for the integer partition function Ivica Martinja 1, and Dragutin Svrtan 1 Department of Physics, University of
More informationMathematics Course 111: Algebra I Part I: Algebraic Structures, Sets and Permutations
Mathematics Course 111: Algebra I Part I: Algebraic Structures, Sets and Permutations D. R. Wilkins Academic Year 1996-7 1 Number Systems and Matrix Algebra Integers The whole numbers 0, ±1, ±2, ±3, ±4,...
More informationPolynomial evaluation and interpolation on special sets of points
Polynomial evaluation and interpolation on special sets of points Alin Bostan and Éric Schost Laboratoire STIX, École polytechnique, 91128 Palaiseau, France Abstract We give complexity estimates for the
More informationChapter 8. P-adic numbers. 8.1 Absolute values
Chapter 8 P-adic numbers Literature: N. Koblitz, p-adic Numbers, p-adic Analysis, and Zeta-Functions, 2nd edition, Graduate Texts in Mathematics 58, Springer Verlag 1984, corrected 2nd printing 1996, Chap.
More information4-Shadows in q-series and the Kimberling Index
4-Shadows in q-series and the Kimberling Index By George E. Andrews May 5, 206 Abstract An elementary method in q-series, the method of 4-shadows, is introduced and applied to several poblems in q-series
More informationMATRIX ALGEBRA AND SYSTEMS OF EQUATIONS. + + x 1 x 2. x n 8 (4) 3 4 2
MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS SYSTEMS OF EQUATIONS AND MATRICES Representation of a linear system The general system of m equations in n unknowns can be written a x + a 2 x 2 + + a n x n b a
More informationMODULAR FORMS ARISING FROM Q(n) AND DYSON S RANK
MODULAR FORMS ARISING FROM Q(n) AND DYSON S RANK MARIA MONKS AND KEN ONO Abstract Let R(w; q) be Dyson s generating function for partition ranks For roots of unity ζ it is known that R(ζ; q) and R(ζ; /q)
More informationA class of trees and its Wiener index.
A class of trees and its Wiener index. Stephan G. Wagner Department of Mathematics Graz University of Technology Steyrergasse 3, A-81 Graz, Austria wagner@finanz.math.tu-graz.ac.at Abstract In this paper,
More informationLecture Notes. Advanced Discrete Structures COT S
Lecture Notes Advanced Discrete Structures COT 4115.001 S15 2015-01-13 Recap Divisibility Prime Number Theorem Euclid s Lemma Fundamental Theorem of Arithmetic Euclidean Algorithm Basic Notions - Section
More informationDefining Valuation Rings
East Carolina University, Greenville, North Carolina, USA June 8, 2018 Outline 1 What? Valuations and Valuation Rings Definability Questions in Number Theory 2 Why? Some Questions and Answers Becoming
More informationA PERIODIC APPROACH TO PLANE PARTITION CONGRUENCES
A PERIODIC APPROACH TO PLANE PARTITION CONGRUENCES MATTHEW S. MIZUHARA, JAMES A. SELLERS, AND HOLLY SWISHER Abstract. Ramanujan s celebrated congruences of the partition function p(n have inspired a vast
More informationNearly Equal Distributions of the Rank and the Crank of Partitions
Nearly Equal Distributions of the Rank and the Crank of Partitions William Y.C. Chen, Kathy Q. Ji and Wenston J.T. Zang Dedicated to Professor Krishna Alladi on the occasion of his sixtieth birthday Abstract
More informationEVALUATION OF A FAMILY OF BINOMIAL DETERMINANTS
EVALUATION OF A FAMILY OF BINOMIAL DETERMINANTS CHARLES HELOU AND JAMES A SELLERS Abstract Motivated by a recent work about finite sequences where the n-th term is bounded by n, we evaluate some classes
More informationBaltic Way 2008 Gdańsk, November 8, 2008
Baltic Way 008 Gdańsk, November 8, 008 Problems and solutions Problem 1. Determine all polynomials p(x) with real coefficients such that p((x + 1) ) = (p(x) + 1) and p(0) = 0. Answer: p(x) = x. Solution:
More informationProof Techniques (Review of Math 271)
Chapter 2 Proof Techniques (Review of Math 271) 2.1 Overview This chapter reviews proof techniques that were probably introduced in Math 271 and that may also have been used in a different way in Phil
More informationMA131 - Analysis 1. Workbook 4 Sequences III
MA3 - Analysis Workbook 4 Sequences III Autumn 2004 Contents 2.3 Roots................................. 2.4 Powers................................. 3 2.5 * Application - Factorials *.....................
More information2 J. ZENG THEOREM 1. In the ring of formal power series of x the following identities hold : (1:4) 1 + X n1 =1 S q [n; ]a x n = 1 ax? aq x 2 b x? +1 x
THE q-stirling NUMBERS CONTINUED FRACTIONS AND THE q-charlier AND q-laguerre POLYNOMIALS By Jiang ZENG Abstract. We give a simple proof of the continued fraction expansions of the ordinary generating functions
More informationCOMBINATORIAL APPLICATIONS OF MÖBIUS INVERSION
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 00, Number 0, Pages 000 000 S 0002-9939(XX)0000-0 COMBINATORIAL APPLICATIONS OF MÖBIUS INVERSION MARIE JAMESON AND ROBERT P. SCHNEIDER (Communicated
More information) = nlog b ( m) ( m) log b ( ) ( ) = log a b ( ) Algebra 2 (1) Semester 2. Exponents and Logarithmic Functions
Exponents and Logarithmic Functions Algebra 2 (1) Semester 2! a. Graph exponential growth functions!!!!!! [7.1]!! - y = ab x for b > 0!! - y = ab x h + k for b > 0!! - exponential growth models:! y = a(
More informationNOTES ON FINITE FIELDS
NOTES ON FINITE FIELDS AARON LANDESMAN CONTENTS 1. Introduction to finite fields 2 2. Definition and constructions of fields 3 2.1. The definition of a field 3 2.2. Constructing field extensions by adjoining
More informationAn Algebraic Identity of F.H. Jackson and its Implications for Partitions.
An Algebraic Identity of F.H. Jackson and its Implications for Partitions. George E. Andrews ( and Richard Lewis (2 ( Department of Mathematics, 28 McAllister Building, Pennsylvania State University, Pennsylvania
More informationarxiv: v1 [math.co] 20 Aug 2015
arxiv:1508.04953v1 [math.co] 20 Aug 2015 On Polynomial Identities for Recursive Sequences Ivica Martinak and Iva Vrsalko Faculty of Science University of Zagreb Bienička cesta 32, HR-10000 Zagreb Croatia
More informationA PARTITION IDENTITY AND THE UNIVERSAL MOCK THETA FUNCTION g 2
A PARTITION IDENTITY AND THE UNIVERSAL MOCK THETA FUNCTION g KATHRIN BRINGMANN, JEREMY LOVEJOY, AND KARL MAHLBURG Abstract. We prove analytic and combinatorial identities reminiscent of Schur s classical
More informationSemester University of Sheffield. School of Mathematics and Statistics
University of Sheffield School of Mathematics and Statistics MAS140: Mathematics (Chemical) MAS15: Civil Engineering Mathematics MAS15: Essential Mathematical Skills & Techniques MAS156: Mathematics (Electrical
More informationSets, Structures, Numbers
Chapter 1 Sets, Structures, Numbers Abstract In this chapter we shall introduce most of the background needed to develop the foundations of mathematical analysis. We start with sets and algebraic structures.
More informationWoon s Tree and Sums over Compositions
2 3 47 6 23 Journal of Integer Sequences, Vol. 2 208, Article 8.3.4 Woon s Tree and Sums over Compositions Christophe Vignat L.S.S., CentraleSupelec Université Paris Sud Orsay, 992 France and Department
More informationDoubly Indexed Infinite Series
The Islamic University of Gaza Deanery of Higher studies Faculty of Science Department of Mathematics Doubly Indexed Infinite Series Presented By Ahed Khaleel Abu ALees Supervisor Professor Eissa D. Habil
More informationHence, the sequence of triangular numbers is given by., the. n th square number, is the sum of the first. S n
Appendix A: The Principle of Mathematical Induction We now present an important deductive method widely used in mathematics: the principle of mathematical induction. First, we provide some historical context
More informationSingular Overpartitions
Singular Overpartitions George E. Andrews Dedicated to the memory of Paul Bateman and Heini Halberstam. Abstract The object in this paper is to present a general theorem for overpartitions analogous to
More informationChapter 11 - Sequences and Series
Calculus and Analytic Geometry II Chapter - Sequences and Series. Sequences Definition. A sequence is a list of numbers written in a definite order, We call a n the general term of the sequence. {a, a
More informationMCR3U Unit 7 Lesson Notes
7.1 Arithmetic Sequences Sequence: An ordered list of numbers identified by a pattern or rule that may stop at some number or continue indefinitely. Ex. 1, 2, 4, 8,... Ex. 3, 7, 11, 15 Term (of a sequence):
More information