where Q denotes an arbitrary operator and C the Hankel path in the complex plane. We obtain [13, 48] and ν$_{q}$ is given by Eq. (42) and (66) for ideal gases and harmonic oscillators, respectively. Next we discuss evaluations by the normal average given by By using Eqs. (B3) and (B4), we obtain ˜ Z$_{q}$ (˜ α ) =      i 2 $_{π}$Γ ( 1 $^{q}$− 1 + 1 ) ∫ $_{C}$( − t ) − 1 / ( $^{q}$− $^{1)}$−$^{1}$e − t Z$_{1}$ [ − ( q − 1)˜ αt ] dt for q > 1 . 0 , 1 Γ[1 / $^{(1}$− q )] ∫ ∞ 0 u 1 / $^{(1}$− q $^{)}$−$^{1}$e − u Z$_{1}$ [(1 − q )˜ αu ] du for q < 1 . 0 , (B11) ˜ Z$_{q}$ (˜ α ) = Tr [1 − ( q − 1)˜ α H ] 1 / ( $^{q}$− $^{1)}$, (B9) ˜ Q$_{q}$ (˜ α ) = 1 ˜ Z$_{q}$ (˜ α ) Tr { [1 − ( q − 1) α H ] 1 / ( $^{q}$− 1) Q } . (B10) Z$_{1}$ ( α ) = Tr e − $^{α H}$, (B7) Q$_{1}$ ( α ) = 1 Z$_{1}$ ( α ) Tr { e − α H Q } (B8) where Z$_{q}$ ( α ) =      1 Γ ( 1 q$_{−}$ $_{1}$) ∫ ∞ 0 u 1 / ( $^{q}$− $^{1)}$−$^{1}$e − $^{u}$Z$_{1}$ [( q − 1) αu ] du for q > 1 . 0 , i 2 $_{π}$Γ ( 1 $^{1}$− q + 1 ) ∫ $_{C}$( − t ) − 1 / $^{(1}$− q $^{)}$−$^{1}$e − $^{t}$Z$_{1}$ [ − (1 − q ) αt ] dt for q < 1 . 0 , (B5) Q$_{q}$ ( α ) =      1 ν$_{q}$ Z$_{q}$ ( α )Γ ( q q$_{−}$ $_{1}$) ∫ ∞ 0 u q/ ( $^{q}$− $^{1)}$−$^{1}$e − $^{u}$Z$_{1}$ [( q − 1) αu ] Q$_{1}$ [( q − 1) αu ] du for q > 1 . 0 , i 2 πν$_{q}$ Z$_{q}$ ( α $_{)}$Γ ( q $^{1}$− q + 1 ) ∫ $_{C}$( − t ) − q/ $^{(1}$− q $^{)}$−$^{1}$e − $^{t}$Z$_{1}$ [ − (1 − q ) αt ] Q$_{1}$ [ − (1 − q ) αt ] dt, for q < 1 . 0 , (B6) 2. The normal average 22