The Multi-variable Arakawa–Kaneko Zeta Function for Non-positive Indices and Its Values at Non-positive Integers

Key Points

• The aim is to define the multi-variable Arakawa-Kaneko zeta function for non-positive indices and explore its properties.
• Defined the multi-variable Arakawa-Kaneko zeta function appropriately.
• Demonstrated analytic continuation to show it behaves as an entire function.
• Explored values at non-positive integers and established a duality relationship.
• Confirmed the function is an entire function through analytic continuation.
• Illustrated that values at non-positive integers demonstrate a unique duality.
• Generalized known results for poly-Bernoulli numbers of C-type.