| Summary: | Since the grand partition function <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>Z</mi><mi>q</mi></msub></semantics></math></inline-formula> for the so-called <i>q</i>-particles (i.e., quons), <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>q</mi><mo>∈</mo><mo>(</mo><mo>−</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo>)</mo></mrow></semantics></math></inline-formula>, cannot be computed by using the standard 2nd quantisation technique involving the full Fock space construction for <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>q</mi><mo>=</mo><mn>0</mn></mrow></semantics></math></inline-formula>, and its <i>q</i>-deformations for the remaining cases, we determine such grand partition functions in order to obtain the natural generalisation of the Plank distribution to <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>q</mi><mo>∈</mo><mo>[</mo><mo>−</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo>]</mo></mrow></semantics></math></inline-formula>. We also note the (non) surprising fact that the right grand partition function concerning the Boltzmann case (i.e., <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>q</mi><mo>=</mo><mn>0</mn></mrow></semantics></math></inline-formula>) can be easily obtained by using the full Fock space 2nd quantisation, by considering the appropriate correction by the Gibbs factor <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>1</mn><mo>/</mo><mi>n</mi><mo>!</mo></mrow></semantics></math></inline-formula> in the <i>n</i> term of the power series expansion with respect to the fugacity <i>z</i>. As an application, we briefly discuss the equations of the state for a gas of free quons or the condensation phenomenon into the ground state, also occurring for the Bose-like quons <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>q</mi><mo>∈</mo><mo>(</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>)</mo></mrow></semantics></math></inline-formula>.
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