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q-Series Identities

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There are a great many beautiful identities involving q-series, some of which follow directly by taking the q-analog of standard combinatorial identities, e.g., the q-binomial theorem

sum_(n=0)^infty((a;q)_nz^n)/((q;q)_n)=((az;q)_infty)/((z;q)_infty)
(1)

(|z|<1, |q|<1; Andrews 1986, p. 10), a special case of an identity due to Euler

(aq;q)_infty=sum_(k=0)^infty((-1)^kq^(k(k+1)/2)a^k)/((q;q)_k)
(2)

(Gasper and Rahman 1990, p. 9; Leininger and Milne 1999), and q-Vandermonde sum

_2phi_1(a,q^(-n);c;q,q)=(a^n(c/a,q)_n)/((c;q)_n),
(3)

where _2phi_1(a,b;c;q,z) is a q-hypergeometric function.

Other q-series identities, e.g., the Jacobi identities, Rogers-Ramanujan identities, and q-hypergeometric identity

_2phi_1(a,b;c;q,z)=((b;q)_infty(az;q)_infty)/((c;q)_infty(z;q)_infty)_2phi_1(c/b,a;az;q,b),
(4)

seem to arise out of the blue. Another such example is

sum_(n=0)^infty((-q;q^2)_nq^(n(n-1))z^n)/((z;q^2)_n)=sum_(n=0)^infty((-zq;q^4)_nq^(n(2n-1))z^n)/((z;q^2)_(2n+1))
(5)

(Gordon and McIntosh 2000).

Hirschhorn (1999) gives the beautiful identity

(q)_infty^5=(q^5)_infty (mod 5)
(6)
=1+4q^5+4q^(10)+q^(25)+q^(35)+4q^(60)+... (mod 5)
(7)

(OEIS A098445). Other modular identities involving the q-series (q)_infty include

(q)_infty^3=sum_(n=0)^(infty)(-1)^n(2n+1)q^(n(n+1)/2) (mod 5)
(8)
=X+2qY (mod 5)
(9)

(Hardy and Wright 1979, Hirschhorn 1999), where

X=product_(n=1)^(infty)(1-q^(25n-15))(1-q^(25n-10))(1-q^(25n))
(10)
=sum_(n=-infty)^(infty)(-1)^nq^((25n^2-5n)/2)
(11)
Y=product_(n=1)^(infty)(1-q^(25n-20))(1-q^(25n-5))(1-q^(25n))
(12)
=sum_(n=-infty)^(infty)(-1)^nq^((25n^2-15n)/2)
(13)

(Hirschhorn 1999).

Zucker (1990) defines the useful notations

(n)=product_(k=1)^(infty)(1-q^(kn))
(14)
=(q^n;q^n)
(15)
[n]=q^(n/24)product_(k=1)^(infty)(1-q^(kn)).
(16)

A set of beautiful identities that can be expressed in this notation were found by M. Trott (pers. comm., Dec. 19, 2000),

0=-5(1)(2)^2(5)^4+4(2)^5(5)(10)+(1)^5(10)^2
(17)
0=-9(1)(2)^4(3)^8+8(2)^9(3)^3(6)+(1)^9(6)^4
(18)
0=-25(1)^2(2)^2(5)^7+16(2)^8(5)(10)^2+5(1)^6(5)^3(10)^2
(19)
+4(1)^5(2)^3(10)^3
(20)
0=-4(1)^2(3)^2(4)^8+3(2)^8(6)^4+(1)^8(4)^2(12)^2
(21)
0=-2(1)^4(4)^(14)+(2)^(14)(8)^4+(1)^8(2)^2(4)^4(8)^4.
(22)

These are closely related to modular equation identities. For example, equation (◇) is an elegant form of Shen (1994) equation (3.12), obtained using the identities

theta_4(q)=((1)^2)/((2))
(23)
theta_4(q^5)=(10)product_(k=1)^(infty)(1-q^(10k-5))^2
(24)
theta_4^3(q^5)=((5)^4)/((10))product_(k=1)^(infty)(1-q^(10k-5))^2
(25)

(OEIS A002448, A089803, and A089804). Similarly, equation (◇) is actually the classical expression

theta_3^2(q)+theta_4^2(q)=2theta_3^2(q^2)
(26)

for the Jacobi theta functions which follows from

theta_3(q)=((2)^5)/((1)^2(4)^2)
(27)
theta_4(q)=((1)^2)/((2))
(28)

(J. Zucker, pers. comm., Nov. 11, 2003).

Another set of identities found by M. Trott (pers. comm., Jul. 8, 2009) are given by

(-1;q)_infty-2(-q;q)_infty=0 q(q^2;q)_infty-(q^2;q)_infty+(q;q)_infty=0 2q(-q^2;q)_infty+2(-q^2;q)_infty-(-1;q)_infty=0 q(-q^2;q)_infty+(-q^2;q)_infty-(-q;q)_infty=0 q(q^(-1);q^2)_infty-q(q;q^2)_infty+(q;q^2)_infty=0 q(-q^(-1);q^2)_infty-q(-q;q^2)_infty-(-q;q^2)_infty=0 (q;q^2)_infty+q(q^3;q^2)_infty-(q^3;q^2)_infty=0 -(q^3;q)_infty+(q^2;q)_infty+q^2(q^3;q)_infty=0 -(-q;q^2)_infty+q(-q^3;q^2)_infty+(-q^3;q^2)_infty=0 (-q^3;q)_infty-(-q^2;q)_infty+q^2(-q^3;q)_infty=0 q(q^(-1);q^3)_infty-q(q^2;q^3)_infty+(q^2;q^3)_infty=0 (q;q^3)_infty+q^2(q^(-2);q^3)_infty-q^2(q;q^3)_infty=0 -(-q;q^3)_infty+q^2(-q^(-2);q^3)_infty-q^2(-q;q^3)_infty=0 q(q^(-1);q^4)_infty-q(q^3;q^4)_infty+(q^3;q^4)_infty=0 q(-q^(-1);q^4)_infty-q(-q^3;q^4)_infty-(-q^3;q^4)_infty=0 (q;q^4)_infty+q^3(q^(-3);q^4)_infty-q^3(q;q^4)_infty=0 -(-q;q^4)_infty+q^3(-q^(-3);q^4)_infty-q^3(-q;q^4)_infty=0 (q^3;q^5)_infty+q^2(q^(-2);q^5)_infty-q^2(q^3;q^5)_infty=0 -(-q^3;q^5)_infty+q^2(-q^(-2);q^5)_infty-q^2(-q^3;q^5)_infty=0 q^3(q^(-3);q^5)_infty+(q^2;q^5)_infty-q^3(q^2;q^5)_infty=0 q^3(-q^(-3);q^5)_infty-(-q^2;q^5)_infty-q^3(-q^2;q^5)_infty=0 (q;q^5)_infty+q^4(q^(-4);q^5)_infty-q^4(q;q^5)_infty=0 -(-q;q^5)_infty+q^4(-q^(-4);q^5)_infty-q^4(-q;q^5)_infty=0 q^3(q^(-3);q^6)_infty-q^3(q^3;q^6)_infty+(q^3;q^6)_infty=0 q^3(-q^(-3);q^6)_infty-q^3(-q^3;q^6)_infty-(-q^3;q^6)_infty=0 q^4(q^(-4);q^6)_infty+(q^2;q^6)_infty-q^4(q^2;q^6)_infty=0 q^4(-q^(-4);q^6)_infty-(-q^2;q^6)_infty-q^4(-q^2;q^6)_infty=0 (q;q^6)_infty+q^5(q^(-5);q^6)_infty-q^5(q;q^6)_infty=0 -(-q;q^6)_infty+q^5(-q^(-5);q^6)_infty-q^5(-q;q^6)_infty=0 q^4(q^(-4);q^7)_infty+(q^3;q^7)_infty-q^4(q^3;q^7)_infty=0 q^4(-q^(-4);q^7)_infty-(-q^3;q^7)_infty-q^4(-q^3;q^7)_infty=0 q^5(q^(-5);q^7)_infty+(q^2;q^7)_infty-q^5(q^2;q^7)_infty=0 q^5(-q^(-5);q^7)_infty-(-q^2;q^7)_infty-q^5(-q^2;q^7)_infty=0 (q;q^7)_infty+q^6(q^(-6);q^7)_infty-q^6(q;q^7)_infty=0 -(-q;q^7)_infty+q^6(-q^(-6);q^7)_infty-q^6(-q;q^7)_infty=0 q^5(q^(-5);q^8)_infty+(q^3;q^8)_infty-q^5(q^3;q^8)_infty=0 q^6(-q^(-6);q^8)_infty-(-q^2;q^8)_infty-q^6(-q^2;q^8)_infty=0 (q;q^8)_infty+q^7(q^(-7);q^8)_infty-q^7(q;q^8)_infty=0 -(-q;q^8)_infty+q^7(-q^(-7);q^8)_infty-q^7(-q;q^8)_infty=0 q^8(-q^(-8);q^(11))_infty-(-q^3;q^(11))_infty-q^8(-q^3;q^(11))_infty=0 -(-q;q^(16))_infty+q^(15)(-q^(-15);q^(16))_infty-q^(15)(-q;q^(16))_infty=0.
(29)

See also

Jacobi identities, q-Hypergeometric Function, q-Series, q-Vandermonde Sum, Ramanujan Theta Functions, Rogers-Ramanujan Identities

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References

Andrews, G. E. q-Series: Their Development and Application in Analysis, Number Theory, Combinatorics, Physics, and Computer Algebra. Providence, RI: Amer. Math. Soc., 1986.Berndt, B. C. "Modular Equations of Degrees 3, 5, and 7 and Associated Theta-Function Identities." Ch. 19 in Ramanujan's Notebooks, Part III. New York:Springer-Verlag, pp. 220-324, 1985.Gasper, G. and Rahman, M. Basic Hypergeometric Series. Cambridge, England: Cambridge University Press, 1990.Gordon, B. and McIntosh, R. J. "Some Eighth Order Mock Theta Functions." J. London Math. Soc. 62, 321-335, 2000.Hardy, G. H. and Wright, E. M. An Introduction to the Theory of Numbers, 5th ed. Oxford, England: Clarendon Press, 1979.Hirschhorn, M. D. "Another Short Proof of Ramanujan's Mod 5 Partition Congruences, and More." Amer. Math. Monthly 106, 580-583, 1999.Leininger, V. E. and Milne, S. C. "Some New Infinite Families of eta-Function Identities." Methods Appl. Anal. 6, 225-248, 1999.Shen, L.-C. "On the Additive Formulae of the Theta Functions and a Collection of Lambert Series Pertaining to the Modular Equations of Degree 5." Trans. Amer. Math. Soc. 345, 323-345, 1994.Sloane, N. J. A. Sequences A002448, A089803, A089804, and A098445 in "The On-Line Encyclopedia of Integer Sequences."Zucker, J. "Further Relations Amongst Infinite Series and Products. II. The Evaluation of Three-Dimensional Lattice Sums." J. Phys. A: Math. Gen. 23, 117-132, 1990.

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q-Series Identities

Cite this as:

Weisstein, Eric W. "q-Series Identities." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/q-SeriesIdentities.html

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