βx:=Γ(x+12)Γ(12)Γ(x+1),βn=(2nn)22n,βn+12=22nπ(n+12)(2nn)
βxr,k:=γr,k(x)βxr
∑n≤kβk+125,−3:=∑0≤k(βk+125,−3−2βk1,−1)−∑k=0n−1βk+125,−3
∑0≤nβn3(8Gπ2+∑k=0n−1βk+122)=∑0≤nβn3∑0≤n(−1)nβn3(8Gπ2+∑k=0n−1βk+122)=12∑0≤n(−1)nβn3∑0≤nβn+123∑k=0nβk4,1=∑0≤nβn3∑0≤n(−1)nβn+123∑k=0nβk4,1=13∑0≤n(−1)nβn3
∑0≤nβn+123,−1=8Gπ2
∑0≤n(βn+124,1−2π1n+1)=6ln2π−28ζ(3)π3∑0≤n(βn+124,1(∑k=0nβk3,−1)2−2π1n+1)=6ln2π
∑0≤n1(n+12)3βn2(∑k=0nβk3,−1)2+∑0<n1n3βn2(∑k=0n−1βk3,−1)2=28ζ(3)∑0≤n(−1)n(n+12)4βn3(∑k=0nβk3,−1)2+∑0<n(−1)n−1n4βn3(∑k=0n−1βk3,−1)2=π4−4π∑0<n1n2∑m=0n−1(−1)mm+12∑0≤n1(n+12)5βn4(∑k=0nβk3,−1)2+∑0<n1n5βn4(∑k=0n−1βk3,−1)2=16π3G−28π2ζ(3)
∑0≤nβn+126,1∑k=0nβk6,1+2∑0≤nβn+124∑k=0nβk4=(∑0≤nβn4)2∑0≤nβn+126,1(∑k=0nβk6,1)(∑k=0nβk3,−1)=(∑0≤nβn4)2−∑0≤nβn+124∑m=0nβm4=∑0≤nβn+126,1(∑k=0nβk6,1∑k≤mβm+123,−1)(∑k=0nβk4,1)
∑0≤nβn4(8Gπ2+∑k=0n−1βk+122)(28ζ(3)π3+∑k=0n−1βk+124,1)=∑0≤nβn6,1(∑n≤kβk+123,−1)3+12∑0≤nβn4∑0≤nβn+126,1(∑k=0nβk3,−1)3=34∑0≤nβn6,1∑0≤nβn+124(∑k=0nβk2)3=14∑0≤nβn6,1∑0≤nβn6,1(8Gπ2+∑k=0n−1βk+122)2=∑0≤nβn+126,1(∑k=0nβk2)4
4∑0≤nβn8,1(28ζ(3)π3+∑k=0n−1βk+124,1)−2∑0≤nβn8,1∑n≤kβk+123,−1=∑0≤nβn8,1(A∑n≤kβk+125,−3+1A∑n≤kβk+125,−1)3∑0≤nβn8,1−∑0≤nβn8,1(∑n≤kβk+123,−1)2=∑0≤nβn8,1(A∑n≤kβk+125,−3−1A∑n≤kβk+125,−1)
A2∑0≤nβn+128,1(∑k=0nβk5,−3)2−1A2∑0≤nβn+128,1(∑k=0nβk5,−1)2=4∑0≤nβn8,1∑n≤kβk+123,−1
3∑0≤nβn8,1(∑n≤kβk+123,−1)2−2∑0≤nβn+128,1(∑k=0nβk3,−1)3=∑0≤nβn8,1∑0≤nβn+128,1(∑k=0nβk3,−1)2+∑0≤nβn8,1(∑n≤kβk+125,−1)(∑n≤kβk+125,−3)=∑0≤nβn8,1∑0≤nβn8,1(∑n≤kβk+123,−1)2+∑0≤nβn+128,1(∑k=0nβk5,−3)(∑k=0nβk5,−1)=∑0≤nβn8,1
∑0≤nβn9,−1(∑n≤kβk+123,−1)3−∑0≤nβn+129,−1(∑k=0nβk3,−1)3=14∑0≤nβn+129,−1(A∑k=0nβk5,−3−1A∑k=0nβk5,−1)2
∑0≤nβn+1210,1(∑k=0nβk4,1)3+∑0≤nβn10,1(A∑n≤kβk+125,−3+1A∑n≤kβk+125,−1)=3∑0≤nβn10,1(28ζ(3)π3+∑k=0n−1βk+124,1)
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