| 1 | """ |
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| 2 | q-expansions of Theta Series |
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| 3 | |
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| 4 | AUTHOR: |
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| 5 | -- William Stein |
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| 6 | """ |
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| 7 | |
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| 8 | from sage.rings.all import Integer, ZZ, PowerSeriesRing |
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| 9 | |
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| 10 | from math import ceil, sqrt |
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| 11 | |
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| 12 | def theta2_qexp(prec=10, var='q', K=ZZ, sparse=False): |
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| 13 | r""" |
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| 14 | Return the $q$-expansion of the series |
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| 15 | $$ |
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| 16 | \theta_2 = \sum_{n odd} q^n. |
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| 17 | $$ |
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| 18 | |
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| 19 | INPUT: |
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| 20 | prec -- integer; the absolute precision of the output |
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| 21 | var -- (default: 'q') variable name |
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| 22 | K -- (default: ZZ) base ring of answer |
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| 23 | |
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| 24 | OUTPUT: |
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| 25 | a power series over K |
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| 26 | |
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| 27 | EXAMPLES: |
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| 28 | sage: theta2_qexp(18) |
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| 29 | q + q^9 + O(q^18) |
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| 30 | sage: theta2_qexp(49) |
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| 31 | q + q^9 + q^25 + O(q^49) |
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| 32 | sage: theta2_qexp(100, 'q', QQ) |
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| 33 | q + q^9 + q^25 + q^49 + q^81 + O(q^100) |
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| 34 | sage: f = theta2_qexp(100, 't', GF(3)); f |
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| 35 | t + t^9 + t^25 + t^49 + t^81 + O(t^100) |
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| 36 | sage: parent(f) |
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| 37 | Power Series Ring in t over Finite Field of size 3 |
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| 38 | sage: theta2_qexp(200) |
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| 39 | q + q^9 + q^25 + q^49 + q^81 + q^121 + q^169 + O(q^200) |
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| 40 | sage: f = theta2_qexp(20,sparse=True); f |
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| 41 | q + q^9 + O(q^20) |
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| 42 | sage: parent(f) |
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| 43 | Sparse Power Series Ring in q over Integer Ring |
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| 44 | """ |
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| 45 | prec = Integer(prec) |
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| 46 | if prec <= 0: |
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| 47 | raise ValueError, "prec must be positive" |
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| 48 | if sparse: |
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| 49 | v = {} |
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| 50 | else: |
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| 51 | v = [Integer(0)] * prec |
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| 52 | one = Integer(1) |
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| 53 | n = int(sqrt(prec)) |
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| 54 | if n*n < prec: |
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| 55 | n += 1 |
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| 56 | for m in xrange(1, n, 2): |
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| 57 | v[m*m] = one |
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| 58 | R = PowerSeriesRing(K, sparse=sparse, names=var) |
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| 59 | return R(v, prec=prec) |
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| 60 | |
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| 61 | def theta_qexp(prec=10, var='q', K=ZZ, sparse=False): |
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| 62 | r""" |
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| 63 | Return the $q$-expansion of the standard $\theta$ series |
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| 64 | $$ |
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| 65 | \theta = 1 + 2\sum_{n=1}{^\infty} q^{n^2}. |
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| 66 | $$ |
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| 67 | |
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| 68 | INPUT: |
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| 69 | prec -- integer; the absolute precision of the output |
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| 70 | var -- (default: 'q') variable name |
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| 71 | K -- (default: ZZ) base ring of answer |
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| 72 | |
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| 73 | OUTPUT: |
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| 74 | a power series over K |
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| 75 | |
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| 76 | EXAMPLES: |
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| 77 | sage: theta_qexp(25) |
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| 78 | 1 + 2*q + 2*q^4 + 2*q^9 + 2*q^16 + O(q^25) |
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| 79 | sage: theta_qexp(10) |
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| 80 | 1 + 2*q + 2*q^4 + 2*q^9 + O(q^10) |
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| 81 | sage: theta_qexp(100) |
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| 82 | 1 + 2*q + 2*q^4 + 2*q^9 + 2*q^16 + 2*q^25 + 2*q^36 + 2*q^49 + 2*q^64 + 2*q^81 + O(q^100) |
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| 83 | sage: theta_qexp(100, 't') |
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| 84 | 1 + 2*t + 2*t^4 + 2*t^9 + 2*t^16 + 2*t^25 + 2*t^36 + 2*t^49 + 2*t^64 + 2*t^81 + O(t^100) |
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| 85 | sage: theta_qexp(100, 't', GF(2)) |
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| 86 | 1 + O(t^100) |
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| 87 | sage: f = theta_qexp(20,sparse=True); f |
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| 88 | 1 + 2*q + 2*q^4 + 2*q^9 + 2*q^16 + O(q^20) |
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| 89 | sage: parent(f) |
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| 90 | Sparse Power Series Ring in q over Integer Ring |
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| 91 | |
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| 92 | """ |
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| 93 | prec = Integer(prec) |
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| 94 | if prec <= 0: |
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| 95 | raise ValueError, "prec must be positive" |
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| 96 | if sparse: |
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| 97 | v = {} |
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| 98 | else: |
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| 99 | v = [Integer(0)] * prec |
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| 100 | v[0] = Integer(1) |
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| 101 | two = Integer(2) |
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| 102 | n = int(sqrt(prec)) |
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| 103 | if n*n != prec: |
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| 104 | n += 1 |
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| 105 | for m in xrange(1, n): |
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| 106 | v[m*m] = two |
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| 107 | |
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| 108 | R = PowerSeriesRing(K, sparse=sparse, names=var) |
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| 109 | return R(v, prec=prec) |
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| 110 | |
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