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A051862
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Perturbation expansion in quantum field theory: scalar case in 6 spacetime dimensions.
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2
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0, 1, 11, 376, 20241, 1427156, 121639250, 12007003824, 1337583507153, 165328009728652, 22404009743110566, 3299256277254713760, 524366465815117346250, 89448728780073829991976, 16301356287284530869810308, 3161258841758986060906197536, 650090787950164885954804021185
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refs;
listen;
history;
text;
internal format)
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OFFSET
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0,3
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LINKS
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FORMULA
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The generating procedure is described by Broadhurst and Kreimer.
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EXAMPLE
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a(31) = 7632236320181399967333968684399053053157812979126909028545984868160 was computed using Kreimer's Hopf algebra of rooted trees. It subsumes 2.6*10^21 terms in quantum field theory.
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PROG
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(Sage)
t = PowerSeriesRing(QQ, 't').gen()
def shadok(c):
"""
fixed point procedure after G. Dunne talk at Kreimer's fest 2020
"""
aa_sur_c = 2 * t * c.derivative() - c - 3
aa = c * aa_sur_c
bb_sur_c = 2 * t * aa.derivative() - aa - 2 * aa_sur_c
bb = c * bb_sur_c
cc_sur_c = 2 * t * bb.derivative() - bb - bb_sur_c
return 3 * t / cc_sur_c
C = (-t / 2).O(2)
for k in range(10):
C = shadok(C)
list(1 / 6 * C(-12 * t))
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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