Remarks on Homogeneous Al-Salam and Carlitz Polynomials

Abstract

Several multilinear generating functions of the homogeneous Al-Salam and Carlitz polynomials are derived from 𝑞-operator. In addition, two interesting relationships of product of this kind of polynomials are obtained.

1. Introduction

The Al-Salam and Carlitz polynomials 𝜙_𝑛^(𝑎)(𝑥)=^𝑛_𝑘=0𝑛𝑘𝑎;𝑞_𝑘𝑥^𝑘 (1) have been studied by many researchers for a long time. The history of these polynomials may go back to Al-Salam and Carlitz in 1965. Since then, these polynomials have been studied by many mathematicians [1–14].

Recently, Cao [7] used Carlitz’s 𝑞-operators to study the following homogeneous Al-Salam and Carlitz polynomials: 𝜙_𝑛^(𝑎)𝑥,𝑦=^𝑛_𝑘=0𝑛𝑘𝑎;𝑞_𝑘𝑥^𝑘𝑦^𝑛−𝑘,𝑦≠0 (2) and he gave some linear generating functions of them. In this paper, we will research these polynomials by some construction of 𝑞-operator. With this method, some new multilinear generating functions can be easily derived. Firstly, we give the following three results which originated from the results about 𝜙_𝑛^(𝑎)(𝑥) which appeared in [1, 2, 5–7, 9, 10].

Theorem 1. If 𝑚,𝑛∈𝑁, max{|𝑥𝑋𝑧|,|𝑥𝑌𝑧|,|𝑦𝑋𝑧|,|𝑦𝑌𝑧|}<1, then ^∞_𝑛=0𝜙^(𝑎)_𝑚+𝑛𝑥,𝑦𝜙_𝑛^(𝑏)()𝑧𝑋,𝑌^𝑛𝑞;𝑞_𝑛=𝑦^𝑚𝑞;𝑞_𝑚𝑏𝑦𝑋𝑧,𝑎𝑥𝑋𝑧;𝑞_∞𝑦𝑌𝑧,𝑦𝑋𝑧,𝑥𝑋𝑧;𝑞_∞×^∞_𝑘=0𝑥/𝑦^𝑘𝑎,𝑦𝑌𝑧,𝑦𝑋𝑧;𝑞_𝑘𝑞,𝑏𝑦𝑋𝑧,𝑎𝑥𝑋𝑧;𝑞_𝑘𝑞;𝑞_𝑚−𝑘×₃Φ₂𝑎𝑞^𝑘,𝑦𝑋𝑧𝑞^𝑘,𝑏𝑋𝑌𝑏𝑦𝑋𝑧𝑞^𝑘,𝑎𝑥𝑋𝑧𝑞^𝑘;𝑞,𝑥𝑌𝑧. (3)

Theorem 2. If 𝑚,𝑛∈𝑁, max{|𝑥𝑡|,|𝑦𝑡|,|𝑋𝑧|,|𝑌𝑧|,|𝑦𝑌𝛾|,|𝑦𝑋𝛾|,|𝑥𝑌𝛾|} <1, then ^∞_{𝑚,𝑛,𝑘=0}𝜙^(𝑎)_𝑚+𝑘𝑥,𝑦𝜙^(𝑏)_𝑛+𝑘()𝑡𝑋,𝑌^𝑚𝑧^𝑛𝛾^𝑘𝑞;𝑞_𝑚𝑞;𝑞_𝑛𝑞;𝑞_𝑘=𝑏𝑋𝑧,𝑎𝑥𝑌𝛾;𝑞_∞𝑌𝑧,𝑋𝑧,𝑦𝑡,𝑦𝑌𝛾,𝑥𝑌𝛾;𝑞_∞×^∞_𝑚=0𝑦𝑋𝛾^𝑚𝑏,𝑌𝑧;𝑞_𝑚𝑏𝑋𝑧;𝑞_𝑚×^∞_𝑘=0𝑥/𝑦^𝑘𝑎,𝑦𝑡,𝑦𝑌𝛾;𝑞_𝑘𝑞,𝑎𝑥𝑌𝛾;𝑞_𝑘𝑞;𝑞_𝑚−𝑘×₂Φ₁𝑎𝑞^𝑘,𝑦𝑌𝛾𝑞^𝑘𝑎𝑥𝑌𝛾𝑞^𝑘;𝑞,𝑥𝑡. (4)

Theorem 3 (cf. [7, Equation (1.9)]). If 𝑘,𝑚,𝑛∈𝑁, max{|𝑥𝑌𝑧|,|𝑥𝑋𝑧|,|𝑋𝑦𝑧|,|𝑌𝑦𝑧|}<1, then ^∞_𝑘=0𝜙^(𝑎)_𝑛+𝑘𝑥,𝑦𝜙^(𝑏)_𝑚+𝑘()𝑧𝑋,𝑌^𝑘𝑞;𝑞_𝑘=𝑦^𝑛𝑌^𝑚𝑞;𝑞_𝑛𝑞;𝑞_𝑚𝑦𝑏𝑋𝑧,𝑥𝑎𝑋𝑧;𝑞_∞𝑦𝑌𝑧,𝑦𝑋𝑧,𝑥𝑋𝑧;𝑞_∞×^∞_𝑘=0(𝑋/𝑌)^𝑘𝑏,𝑦𝑌𝑧;𝑞_𝑘𝑞,𝑦𝑏𝑋𝑧;𝑞_𝑘𝑞;𝑞_𝑚−𝑘×^∞_𝑗=0𝑥/𝑦^𝑗𝑎,𝑦𝑌𝑧𝑞^𝑘,𝑦𝑋𝑧;𝑞_𝑗𝑞,𝑦𝑏𝑋𝑧𝑞^𝑘,𝑥𝑎𝑋𝑧;𝑞_𝑗𝑞;𝑞_𝑛−𝑗×₃Φ₂𝑎𝑞^𝑗,𝑦𝑋𝑧𝑞^𝑗,𝑏𝑋𝑌𝑏𝑦𝑋𝑧𝑞^𝑘+𝑗,𝑎𝑥𝑋𝑧𝑞^𝑗;𝑞,𝑥𝑌𝑧𝑞^𝑘. (5)

Now further using our method, we can deduce more results of multilinear generating functions.

Theorem 4. If 𝐺∈𝑍⁺, 𝑎₁=𝑞^−𝐺, |𝑎₁𝑥₁/𝑦₁|<1, then ^∞_{𝑚1,𝑚2,𝑚3=0}𝜙^(𝑎1)_{𝑚1+𝑚2+𝑚3}𝑥₁,𝑦₁𝜙^(𝑎2)_𝑚1+𝑚3𝑥₂,𝑦₂×𝜙^(𝑎3)_𝑚1+𝑚2𝑥₃,𝑦₃𝑡^𝑚1₁𝑡^𝑚2₂𝑡^𝑚3₃𝑞;𝑞_𝑚1𝑞;𝑞_𝑚2𝑞;𝑞_𝑚3=𝑎₁𝑥₁/𝑦₁,𝑎₃𝑥₃𝑡₂𝑦₁,𝑎₂𝑥₂𝑡₁𝑦₁𝑦₃;𝑞_∞𝑥₁/𝑦₁,𝑡₂𝑦₁𝑦₃,𝑡₃𝑦₁𝑦₂,𝑡₁𝑦₁𝑦₂𝑦₃,𝑥₃𝑡₂𝑦₁,𝑥₂𝑡₁𝑦₁𝑦₃;𝑞_∞×^∞_𝑗1=0𝑎₃;𝑞_𝑗1𝑥₃𝑡₁𝑦₁𝑦₂^𝑗1𝑞;𝑞_𝑗1𝑞;𝑞_𝑗1×^∞_𝑗2=0𝑥₂/𝑦₂^𝑗2𝑎₂;𝑞_𝑗2𝑞;𝑞_𝑗2𝑞;𝑞_{𝑗1−𝑗2}^∞_𝑗3=0𝑥₂𝑡₃𝑦₁^𝑗3𝑎₂𝑞^𝑗2;𝑞_𝑗3𝑞;𝑞_𝑗3×^∞_𝑗4=0𝑎₁,𝑥₃𝑡₂𝑦₁,𝑥₂𝑡₁𝑦₁𝑦₃;𝑞_𝑗4(−1)^𝑗4𝑞^{−(𝑗4−1)𝑗4/2}𝑞,𝑥₁/𝑦₁𝑞^𝑗4;𝑞_𝑗4×𝑥₁𝑞^𝑗1+𝑗3𝑦₁^𝑗4𝑡₃𝑦₁𝑦₂;𝑞_𝑗2+𝑗4𝑡₂𝑦₁𝑦₃;𝑞_𝑗1+𝑗4𝑎₃𝑥₃𝑡₂𝑦₁;𝑞_𝑗1+𝑗4×𝑡₁𝑦₁𝑦₂𝑦₃;𝑞_{𝑗2+𝑗3+𝑗4}𝑎₂𝑥₂𝑡₁𝑦₁𝑦₃;𝑞_{𝑗2+𝑗3+𝑗4}, (6) provided that max{|𝑡_𝑖𝑥_𝑗𝑦_𝑙|,|𝑡₁𝑥_𝑖𝑦_𝑗𝑦_𝑙|,|𝑡₁𝑦₁𝑦₂𝑦₃|}<1, where 1≤𝑖≠𝑗≠𝑙≤3.

Theorem 5. If 𝐺∈𝑍⁺, 𝑎₁=𝑞^−𝐺, then ^∞_{𝑚1,𝑚2,𝑚3=0}𝜙^(𝑎1)_{𝑚1+𝑚2+𝑚3}𝑥₁,𝑦₁𝜙^(𝑎2)_𝑚1+𝑚2𝑥₂,𝑦₂×𝜙^(𝑎3)_𝑚3𝑥₃,𝑦₃𝑡^𝑚1₁𝑡^𝑚2₂𝑡^𝑚3₃𝑞;𝑞_𝑚1𝑞;𝑞_𝑚2𝑞;𝑞_𝑚3=𝑎₁𝑥₁/𝑦₁,𝑎₃𝑥₃𝑡₃𝑦₁,𝑎₂𝑥₂𝑡₂𝑦₁;𝑞_∞𝑥₁/𝑦₁,𝑡₃𝑦₁𝑦₃,𝑡₃𝑦₁𝑥₃,𝑡₁𝑦₁𝑦₂,𝑡₂𝑦₁𝑦₂,𝑡₂𝑥₂𝑦₁;𝑞_∞×^∞_{𝑗1,𝑗2=0}𝑎₂;𝑞_𝑗1𝑡₁𝑥₂𝑦₁^𝑗1(−1)^𝑗2𝑞^{−(𝑗2−1)𝑗2/2}𝑞;𝑞_𝑗1×𝑎₁,𝑡₃𝑦₃𝑦₁,𝑡₃𝑥₃𝑦₁,𝑡₂𝑥₂𝑦₁,𝑡₁𝑦₂𝑦₁;𝑞_𝑗2𝑞,𝑞𝑥₁/𝑦₁𝑞^𝑗2,𝑎₃𝑡₂𝑥₃𝑦₁;𝑞_𝑗2×𝑡₂𝑦₂𝑦₁;𝑞_𝑗1+𝑗2𝑎₂𝑡₂𝑥₂𝑦₁;𝑞_𝑗1+𝑗2𝑞^𝑗1𝑥₁𝑦₁^𝑗2, (7) provided that max{|𝑦₁𝑦₂𝑡₁|,|𝑦₁𝑦₂𝑡₂|, |𝑦₁𝑦₃𝑡₃|, |𝑥₃𝑦₁𝑡₃|, |𝑥₂𝑦₁𝑡₁|, |𝑥₁𝑦₁𝑡₂|,|𝑎₁𝑥₁/𝑦₁|}<1.

Theorem 6. If 𝑚_𝑖∈𝑁, max{|𝑥₁|,|𝑦₁𝑡_𝑖|,|𝑥₁𝑡_𝑖|}<1, 𝑖=1,2,3, then ^∞_{𝑚1,𝑚2,𝑚3=0}𝜙_𝑚^(𝑎)_{1+𝑚2+𝑚3}𝑥₁,𝑦₁×𝑡^𝑚1₁𝑡^𝑚2₂−𝑡₃^𝑚3𝑞^{𝑚3(𝑚3−1)/2}𝑞;𝑞_𝑚1𝑞;𝑞_𝑚2𝑞;𝑞_𝑚3=𝑦₁𝑡₃,𝑎𝑥₁𝑡₂;𝑞_∞𝑦₁𝑡₁,𝑦₁𝑡₂,𝑥₁𝑡₂;𝑞_∞×₃Φ₂𝑎,𝑦₁𝑡₂,𝑡₃𝑡₁𝑦₁𝑡₃,𝑎𝑥₁𝑡₂;𝑞,𝑥₁𝑡₁. (8)

Theorem 7. If 𝑚_𝑖∈𝑁,𝑛,𝐺∈𝑍⁺, 𝑎=𝑞^−𝐺, max{|𝑦𝑡_𝑖|,|𝑎𝑥/𝑦|}<1, 𝑖=1,2,…,𝑛, then ^∞_{𝑚1,…,𝑚𝑛=0}𝜙_𝑚^(𝑎)_{1+⋯+𝑚𝑛}𝑡𝑥,𝑦^𝑚11⋯𝑡^𝑚𝑛_𝑛𝑞;𝑞_𝑚1⋯𝑞;𝑞_𝑚𝑛=𝑎𝑥/𝑦;𝑞_∞𝑥/𝑦,𝑦𝑡₁,𝑦𝑡₂,…,𝑦𝑡_𝑛;𝑞_∞×^∞_𝑗=0𝑎,𝑦𝑡₁,𝑦𝑡₂,…,𝑦𝑡_𝑛;𝑞_𝑗(−1)^𝑗𝑞^{−(𝑗−1)𝑗/2}𝑞,𝑥/𝑦𝑞^𝑗;𝑞_𝑗𝑥𝑦^𝑗. (9)

Theorem 8. If 𝑚_𝑖∈𝑁, 𝑡_𝑖≠0, max{|𝑥|,|𝑦𝑡_𝑖|,|𝑥𝑡_𝑖|}<1, 𝑖=1,2,3,4, then ^∞_{𝑚1,𝑚2,𝑚3,𝑚4=0}𝜙_𝑚^(𝑎)_{1+𝑚2+𝑚3+𝑚4}×𝑥,𝑦−𝑡₁^𝑚1𝑡^𝑚2₂𝑡^𝑚3₃−𝑡₄^𝑚4𝑞^{𝑚1(𝑚1−1)/2}𝑞^{𝑚4(𝑚4−1)/2}𝑞;𝑞_𝑚1𝑞;𝑞_𝑚2𝑞;𝑞_𝑚3𝑞;𝑞_𝑚4=𝑦𝑡₁,𝑦𝑡₄;𝑞_∞𝑦𝑡₂,𝑦𝑡₃;𝑞_∞^∞_𝑗=0𝑎,𝑡₁/𝑡₂,𝑦𝑡₃;𝑞_𝑗𝑥𝑡₂^𝑗𝑞,𝑦𝑡₁,𝑦𝑡₄;𝑞_𝑗×₂Φ₁𝑎𝑞^𝑗,𝑡₄𝑡₃𝑦𝑡₄𝑞^𝑗;𝑞,𝑥𝑡₃. (10)

Theorem 9. If 𝑚_𝑖∈𝑁, max{|𝑦_𝑖𝑦𝑡_𝑖|,|𝑥_𝑖𝑦𝑡_𝑖|,|𝑎₀/𝑥₀|}<1, 𝑖=0,1,2,3, then ^∞_{𝑚,𝑚1,𝑚2,𝑚3=0}𝜙_𝑚^(𝑎)_{1+𝑚2+𝑚3+𝑠}𝑥,𝑦𝜙^(𝑎)_𝑚+𝑟×𝑥₀,𝑦₀𝜙^(𝑎1)_𝑚1𝑥₁,𝑦₁𝜙^(𝑎2)_𝑚2𝑥₂,𝑦₂×𝜙^(𝑎3)_𝑚3𝑥₃,𝑦₃×𝑡^𝑚𝑡₁^𝑚1𝑡^𝑚2₂𝑡^𝑚3₃𝑞;𝑞_𝑚𝑞;𝑞_𝑚1𝑞;𝑞_𝑚2𝑞;𝑞_𝑚3=𝑦^𝑠𝑦^𝑟₀𝑞;𝑞_𝑟𝑎𝑥𝑦,𝑎₁𝑦𝑥₁𝑡₁,𝑎₂𝑦𝑥₂𝑡₂,𝑎₃𝑦𝑥₃𝑡₃;𝑞_∞×𝑥𝑦,𝑦𝑦₁𝑡₁,𝑦𝑥₁𝑡₁,𝑦𝑦₂𝑡₂,𝑦𝑥₂𝑡₂,𝑦𝑦₃𝑡₃,𝑦𝑥₃𝑡₃,𝑦𝑦₀𝑡;𝑞_∞⁻¹×^∞_{𝑗1,𝑗2=0}𝑥₀/𝑦₀^𝑗1𝑦𝑥₀𝑡^𝑗2𝑎₀;𝑞_𝑗1+𝑗2𝑞;𝑞_𝑗1𝑞;𝑞_𝑗2𝑞;𝑞_𝑟−𝑗1×^∞_𝑗3=0𝑎,𝑦𝑦₁𝑡₁,𝑦𝑥₁𝑡₁,𝑦𝑦₂𝑡₂,𝑦𝑥₂𝑡₂,𝑦𝑦₃𝑡₃,𝑦𝑥₃𝑡₃;𝑞_𝑗3×𝑦𝑦₀𝑡;𝑞_𝑗1+𝑗3×𝑞𝑦𝑥,𝑎₁𝑦𝑥₁𝑡₁,𝑎₂𝑦𝑥₂𝑡₂,𝑎₃𝑦𝑥₃𝑡₃;𝑞_𝑗3⁻¹×𝑞^{𝑗3(𝑗2+𝑠+1)}, (11) where 𝑡₀=𝑡, 𝑚₀=𝑚, and 𝐹∈𝑍⁺,𝑎₀=𝑞^−𝐹.

Theorem 10. If 𝑀_𝑖∈𝑍⁺, 𝑎_𝑖=𝑞^−𝑀𝑖, 𝑥_𝑖≠0, and 𝑦_𝑖≠0, then ^∞_{𝑚1,𝑚2,𝑚3,𝑚4=0}𝜙^(𝑎1)_{𝑚2+𝑚3+𝑚4}𝑥₁,𝑦₁×𝜙^(𝑎2)_{1+𝑚3+𝑚4}𝑥₂,𝑦₂𝜙^(𝑎3)_{𝑚1+𝑚2+𝑚4}×𝑥₃,𝑦₃𝜙^(𝑎4)_{𝑚1+𝑚2+𝑚3}𝑥₄,𝑦₄×−𝑡₁^𝑚1𝑡^𝑚2₂𝑡^𝑚3₃−𝑡₄^𝑚4𝑞^{𝑚1(𝑚1−1)/2}𝑞^{𝑚4(𝑚4−1)/2}𝑞;𝑞_𝑚1𝑞;𝑞_𝑚2𝑞;𝑞_𝑚3𝑞;𝑞_𝑚4=𝑎₄𝑥₄𝑦₄,𝑎₃𝑥₃𝑦₃,𝑎₂𝑥₂𝑦₂,𝑎₁𝑥₁𝑦₁,𝑡₁𝑦₂𝑦₃𝑦₄,𝑡₄𝑦₁𝑦₂𝑦₃;𝑞_∞×𝑥₄𝑦₄,𝑥₃𝑦₃,𝑥₂𝑦₂,𝑥₁𝑦₁,𝑡₂𝑦₁𝑦₃𝑦₄,𝑡₃𝑦₁𝑦₂𝑦₄;𝑞_∞⁻¹×^∞_{𝑘1,𝑘2,𝑘3,𝑘4=0}𝑎₄;𝑞_𝑘4𝑎₃;𝑞_𝑘3𝑎₂;𝑞_𝑘2×𝑎₁;𝑞_𝑘1𝑞^{𝑘1+𝑘2+𝑘3+𝑘4}×𝑞,𝑞𝑦₄𝑥₄;𝑞_𝑘4𝑞,𝑞𝑦₃𝑥₃;𝑞_𝑘3×𝑞,𝑞𝑦₂𝑥₂;𝑞_𝑘2𝑞,𝑞𝑦₁𝑥₁;𝑞_𝑘1⁻¹×𝑡₁𝑦₂𝑦₃𝑦₄;𝑞_{𝑘2+𝑘3+𝑘4}𝑡₄𝑦₁𝑦₂𝑦₃;𝑞_{𝑘1+𝑘2+𝑘3}𝑡₂𝑦₁𝑦₃𝑦₄;𝑞_{𝑘1+𝑘3+𝑘4}𝑡₃𝑦₁𝑦₂𝑦₄;𝑞_{𝑘1+𝑘2+𝑘4}, (12) provided that max{|𝑎_𝑖𝑥_𝑖/𝑦_𝑖|,|𝑡_𝑖𝑦_𝑗𝑦_𝑘𝑦_𝑙|}<1, where 1≤𝑖≠𝑗≠𝑘≠𝑙≤4.

Polynomials (2) evidently reduce to the Rogers-Szegö polynomials (cf. [9]) 𝜙_𝑛⁽⁰⁾(𝑥)=𝐻_𝑛(𝑥)=^𝑛_𝑘=0𝑛𝑘𝑥^𝑘 (13) when 𝑎=0 and 𝑦=1. And when 𝑦=1 they reduce to the common Al-Salam and Carlitz polynomials (1) (cf. [9]).

So now we take some special cases for checking.

Let 𝑎=𝑏=0, 𝑥=𝑌=1 in (3); we have the following.

Corollary 11 (cf. [6, Theorem 1.1] or [9, Equation (4.1)]). If 𝑚,𝑛∈𝑁, then ^∞_𝑛=0𝐻_𝑚+𝑛𝑦𝐻_𝑛(𝑋)𝑧^𝑛𝑞;𝑞_𝑛=𝑦^𝑚𝑦𝑋𝑧²;𝑞_∞𝑧,𝑋𝑧,𝑦𝑧,𝑦𝑋;𝑞_∞×₃Φ₁𝑞^−𝑚,𝑦𝑧,𝑦𝑋𝑧𝑦𝑋𝑧²𝑞;𝑞,^𝑚𝑦, (14) provided that max{|𝑧|,|𝑦𝑧|,|𝑋𝑧|,|𝑦𝑋𝑧|}<1.

Remark 12 (from [6, Equation (3.1)]). We know that Corollary 11 is equivalent to Theorem 1.1 given in [6]. And if we take 𝑚=0 and 𝑦=𝑌=1, (3) turns to [9, Equation (1.2)].

Taking 𝑎=𝑏=0,𝑥=𝑌=1 in (4), we have the following.

Corollary 13 (cf. [6, Theorem 1.2] or [9, Equation (1.3)]). If 𝑚,𝑛∈𝑁, then ^∞_{𝑚,𝑛,𝑘=0}𝐻_𝑚+𝑘𝑦𝐻_𝑛+𝑘(𝑋)𝑡^𝑚𝑧^𝑛𝛾^𝑘𝑞;𝑞_𝑚𝑞;𝑞_𝑛𝑞;𝑞_𝑘=𝑦𝑡𝛾,𝑦𝑋𝑧𝛾;𝑞_∞𝑡,𝑧,𝑋𝑧,𝑦𝑡,𝛾,𝑦𝛾,𝑦𝑋𝛾;𝑞_∞×₃Φ₂,𝑧,𝑦𝑡,𝑦𝛾𝑦𝑡𝛾,𝑦𝑋𝑧𝛾;𝑞,𝑋𝛾 (15) provided that max{|𝑡|,|𝑧|,|𝛾|,|𝑦𝑡|,|𝑦𝛾|,|𝑋𝑧|,|𝑋𝛾|,|𝑦𝑋𝛾|}<1.

Remark 14. Using Hall’s transformation [15, Equation (3.2.7)] ₃Φ₂𝑎,𝑏,𝑐𝑑,𝑒;𝑞,𝑑𝑒()=𝑎𝑏𝑐𝑒/𝑎,𝑑𝑒/𝑏𝑐;𝑞_∞𝑒,𝑑𝑒/𝑎𝑏𝑐;𝑞_∞³Φ₂𝑑𝑎,𝑏,𝑑𝑐𝑑,𝑑𝑒()𝑒𝑏𝑐;𝑞,𝑎, (16) we find that Corollary 13 is equivalent to Theorem 1.2 given in [6]. And if we take 𝑦=𝑌=1, then, with simplifying, (4) turns to [9, Equation (3.5)].

Taking 𝑎=𝑏=0 and 𝑥=𝑋=1 in (5), we have the following.

Corollary 15 (cf. [6, Theorem 1.3] or [9, Equation (1.4)]). If 𝑚,𝑛,𝑘∈𝑁, then ^∞_𝑘=0𝐻_𝑛+𝑘𝑦𝐻_𝑚+𝑘(𝑌)𝑧^𝑘𝑞;𝑞_𝑘=𝑦𝑌𝑧²;𝑞_∞𝑧,𝑦𝑧,𝑌𝑧,𝑦𝑌𝑧;𝑞_∞×^𝑚𝑛_𝑘=0𝑗=0𝑚𝑘𝑚𝑗𝑌^𝑚−𝑘𝑦^𝑛−𝑗𝑦𝑌𝑧;𝑞_𝑘+𝑗𝑦𝑌𝑧²;𝑞_𝑘+𝑗×𝑌𝑧;𝑞_𝑘𝑦𝑧;𝑞_𝑗, (17) provided that max{|𝑧|,|𝑦𝑧|,|𝑌𝑧|,|𝑦𝑌𝑧|}<1.

The rest of this paper is organized as follows. In Section 2, we will give some notations and lemmas. In Section 3, we give the proofs of theorems. Section 4 describes the relationship between Ψ_𝑛(𝑎,𝑏;𝑥₁,𝑦₁;𝑧) (cf. [4, Equation (2.2)]) and 𝜙_𝑛^(𝑎)(𝑥,𝑦). In addition, an interesting relationship between the polynomial multiplications is given.

2. Notations and Some Lemmas

In this paper, we apply the standard notations that follow from [15] and assume that 0<|𝑞|<1; the 𝑞-shifted factorial and its compact factorials are defined, respectively, by 𝑎;𝑞₀=1;𝑎;𝑞_𝑛=^𝑛−1_𝑘=01−𝑎𝑞^𝑘,𝑎;𝑞_∞=^∞_𝑘=01−𝑎𝑞^𝑘 (18) and (𝑎₁,…,𝑎_𝑚;𝑞)_𝑛=(𝑎₁;𝑞)_𝑛,…,(𝑎_𝑚;𝑞)_𝑛, where 𝑛=0,1,…,∞. And we use 𝑁 to denote the set of nonnegative integers and 𝑍⁺ to denote the set of positive integers.

For any complex number 𝛼, we have 𝑎;𝑞_𝛼=𝑎;𝑞_∞𝑎𝑞^𝛼;𝑞_∞. (19)

The 𝑞-binomial coefficient and the 𝑞-binomial theorem are given by 𝑛𝑘=𝑞;𝑞_𝑛𝑞;𝑞_𝑘𝑞;𝑞_𝑛−𝑘,^∞_𝑚=0𝑎;𝑞_𝑚𝑞;𝑞_𝑚𝑦^𝑚=𝑎𝑦;𝑞_∞𝑦;𝑞_∞, (20) respectively.

The basic hypergeometric series _𝑟Φ_𝑠 is defined by _𝑟Φ_𝑠𝑎₁,…,𝑎_𝑟𝑏₁,…,𝑏_𝑠=;𝑞,𝑥_∞𝑛=0𝑎₁,𝑎₂,…,𝑎_𝑟;𝑞_𝑛𝑞,𝑏₁,…,𝑏_𝑠;𝑞_𝑛×(−1)^𝑛𝑞^{𝑛(𝑛−1)/21+𝑠−𝑟}𝑥^𝑛. (21)

The 𝑞-derivative operator 𝐷_𝑞 and 𝑞-shifted operator 𝜂 (cf. [1, 6, 7, 9, 10, 16–23]), acting on the variable 𝑥, are defined by 𝐷_𝑞𝑓(𝑥=𝑓(𝑥))−𝑓𝑥𝑞𝑥,𝜂𝑓(𝑥)=𝑓𝑥𝑞. (22)

The 𝑞-Leibnitz rule for the product of two functions (cf. [17–25]) is given by 𝐷^𝑛_𝑞𝑓(𝑥)𝑔(𝑥)=^𝑛_𝑘=0𝑛𝑘𝑞^{𝑘(𝑘−𝑛)}𝐷^𝑘_𝑞𝑓(𝑥)𝐷_𝑞^𝑛−𝑘𝑔𝑥𝑞^𝑘. (23)

From definition, we can easily obtain 𝐷_𝑞=𝑥⁻¹(1−𝜂); by mathematical induction, we have the following proposition.

Proposition 16 (cf. [20, Equation (6)]). If 𝑓(𝑥) is any analytical function, then 𝐷^𝑛_𝑞𝑓(𝑥)=𝑥^𝑛−𝑛_𝑘=0𝑞^−𝑛;𝑞_𝑘𝑞;𝑞_𝑘𝑞𝜂^𝑘𝑓(𝑥). (24)

In [19, 20], we had established the following 𝑞-operator structure: ₁Φ₀𝑏−;𝑞,𝑐𝐷_𝑞=^∞_𝑛=0𝑏;𝑞_𝑛𝑐𝐷_𝑞^𝑛𝑞;𝑞_𝑛. (25) For convenient, we use 𝐹(𝑎;𝑥𝐷_𝑞)_𝑦 to denote the operator ₁Φ₀^𝑎₋;𝑞,𝑥𝐷_𝑞 acting on variable 𝑦. From this definition we can easily get the following Lemma.

Lemma 17. One has 𝜙_𝑛^(𝑎)𝑥,𝑦=𝐹𝑎;𝑥𝐷_𝑞𝑦𝑦^𝑛. (26)

Proof. For 𝐷_𝑞^(𝑘)𝑦^𝑛=𝑞;𝑞_𝑛𝑞;𝑞_𝑛−𝑘𝑦^𝑛−𝑘, (27) we have 𝐹𝑎;𝑥𝐷_𝑞𝑦𝑦^𝑛=^∞_𝑘=0𝑎;𝑞_𝑘𝑥^𝑘𝑞;𝑞_𝑘𝑞;𝑞_𝑛𝑞;𝑞_𝑛−𝑘𝑦^𝑛−𝑘=^𝑛_𝑘=0𝑛𝑘𝑎;𝑞_𝑘𝑥^𝑘𝑦^𝑛−𝑘, (28) which completes the proof.

It is the reason why we employ this operator to study the properties of homogeneous Al-Salam and Carlitz polynomials.

From Proposition 16, we may easily obtain the following identity.

Lemma 18. If 𝐺∈𝑍⁺, 𝑎=𝑞^−𝐺, |𝑎𝑥/𝑦|<1 and 𝑓(𝑥) is any analytical function, then 𝐹𝑎;𝑥𝐷_𝑞𝑦𝑓𝑦=𝑎𝑥/𝑦;𝑞_∞𝑥/𝑦;𝑞_∞^∞_𝑘=0(−1)^𝑘𝑎;𝑞_𝑘𝑥/𝑦^𝑘𝑞,𝑥/𝑦𝑞^𝑘;𝑞_𝑘×𝑞^{−𝑘(𝑘−1)/2}𝑓𝑦𝑞^𝑘. (29)

Proof. Applying Proposition 16, LHS of Lemma 18 comes to ^∞_𝑛=0𝑎;𝑞_𝑛𝑥/𝑦^𝑛𝑞;𝑞_𝑛×^𝑛_𝑘=0𝑞;𝑞_𝑛𝑞^𝑘𝑞;𝑞_𝑘𝑞;𝑞_𝑛−𝑘(−1)^𝑘𝑞^{𝑘(𝑘−1)/2−𝑛𝑘}𝑓𝑦𝑞^𝑘=^∞_𝑘=0(−1)^𝑘𝑎;𝑞_𝑘𝑥/𝑦^𝑘𝑞;𝑞_𝑘𝑞^{−𝑘(𝑘−1)/2}𝑓𝑦𝑞^𝑘×^∞_𝑛=0𝑎𝑞^𝑘;𝑞_𝑛𝑞;𝑞_𝑛𝑥𝑦𝑞^𝑘𝑛=^∞_𝑘=0(−1)^𝑘𝑎;𝑞_𝑘𝑥/𝑦^𝑘𝑞;𝑞_𝑘𝑞^{−𝑘(𝑘−1)/2}𝑓𝑦𝑞^𝑘×𝑎𝑥/𝑦;𝑞_∞𝑥/𝑦𝑞^𝑘;𝑞_∞=𝑎𝑥/𝑦;𝑞_∞𝑥/𝑦;𝑞_∞^∞_𝑘=0(−1)^𝑘𝑎;𝑞_𝑘𝑥/𝑦^𝑘𝑞,𝑥/𝑦𝑞^𝑘;𝑞_𝑘×𝑞^{−𝑘(𝑘−1)/2}𝑓𝑦𝑞^𝑘. (30)
We complete the proof.

Remark 19. To calculate the inner sum of the above, under the condition 𝑎=𝑞^−𝐺, 𝑞-binomial theorem is usable.

Lemma 20 (cf. [20, Lemma 1]). If |𝑠𝑐|<1, |𝑡𝑐|<1, then 𝐹𝑏;𝑐𝐷_𝑞𝑎𝑎𝜔;𝑞_∞𝑎𝑠,𝑎𝑡;𝑞_∞=𝑎𝜔,𝑏𝑐𝑡;𝑞_∞𝑎𝑠,𝑎𝑡,𝑐𝑡;𝑞_∞³Φ₂𝜔𝑏,𝑎𝑡,𝑠.𝑎𝜔,𝑏𝑐𝑡;𝑞,𝑠𝑐 (31)

Proof. LHS of Lemma 20 equates to ^∞_𝑛=0𝑏;𝑞_𝑛𝑐^𝑛𝑞;𝑞_𝑛^𝑛_𝑘=0𝑛𝑘𝑞^{𝑘(𝑘−𝑛)}𝐷^𝑘_𝑞𝑎𝜔;𝑞_∞𝑎𝑠;𝑞_∞×𝐷_𝑞^𝑛−𝑘1𝑎𝑡𝑞^𝑘;𝑞_∞=^∞_𝑛=0𝑏;𝑞_𝑛𝑐^𝑛𝑞;𝑞_𝑛×^𝑛_𝑘=0𝑛𝑘𝑞^{𝑘(𝑘−𝑛)}𝑠^𝑘𝜔/𝑠;𝑞_𝑘𝑎𝜔𝑞^𝑘;𝑞_∞𝑎𝑠;𝑞_∞𝑡𝑞^{𝑘𝑛−𝑘}𝑎𝑡𝑞^𝑘;𝑞_∞=𝑎𝜔;𝑞_∞𝑎𝑠,𝑎𝑡;𝑞_∞^∞_𝑘=0𝑎𝑡,𝜔/𝑠;𝑞_𝑘𝑞,𝑎𝜔;𝑞_𝑘𝑠^𝑘×^∞_𝑛=𝑘𝑏;𝑞_𝑛𝑞;𝑞_𝑛−𝑘𝑐^𝑛𝑡^𝑛−𝑘=𝑎𝜔;𝑞_∞𝑎𝑠,𝑎𝑡;𝑞_∞^∞_𝑘=0𝑎𝑡,𝜔/𝑠;𝑞_𝑘𝑞,𝑎𝜔;𝑞_𝑘(𝑠𝑐)^𝑘×^∞_𝑚=0𝑏;𝑞_𝑚+𝑘𝑞;𝑞_𝑚(𝑡𝑐)^𝑚=𝑎𝜔;𝑞_∞𝑎𝑠,𝑎𝑡;𝑞_∞^∞_𝑘=0𝑎𝑡,𝜔/𝑠,𝑏;𝑞_𝑘𝑞,𝑎𝜔;𝑞_𝑘(𝑠𝑐)^𝑘×^∞_𝑚=0𝑏𝑞^𝑘;𝑞_𝑚𝑞;𝑞_𝑚(𝑡𝑐)^𝑚=𝑎𝜔;𝑞_∞𝑎𝑠,𝑎𝑡;𝑞_∞^∞_𝑘=0𝑎𝑡,𝜔/𝑠,𝑏;𝑞_𝑘𝑞,𝑎𝜔;𝑞_𝑘(𝑠𝑐)^𝑘𝑏𝑐𝑡𝑞^𝑘;𝑞_∞𝑐𝑡;𝑞_∞=𝑎𝜔,𝑏𝑐𝑡;𝑞_∞𝑎𝑠,𝑎𝑡,𝑐𝑡;𝑞_∞³Φ₂𝜔𝑏,𝑎𝑡,𝑠.𝑎𝜔,𝑏𝑐𝑡;𝑞,𝑠𝑐 (32) We complete the proof.

The special cases of Lemma 20 as taking 𝑡=0, 𝜔=𝑡=0, and 𝜔=0, respectively, will be frequently used.

Lemma 21. For |𝑠𝑐|<1, 𝐹𝑏;𝑐𝐷_𝑞𝑎𝑎𝜔;𝑞_∞𝑎𝑠;𝑞_∞=𝑎𝜔;𝑞_∞𝑎𝑠;𝑞_∞²Φ₁𝜔𝑏,𝑠.𝑎𝜔;𝑞,𝑠𝑐 (33)

Lemma 22. If |𝑠𝑐|<1, then 𝐹𝑏;𝑐𝐷_𝑞𝑎1𝑎𝑠;𝑞_∞=𝑏𝑐𝑠;𝑞_∞𝑎𝑠,𝑐𝑠;𝑞_∞. (34)

Lemma 23. For |𝑠𝑐|<1, |𝑡𝑐|<1, 𝐹𝑏;𝑐𝐷_𝑞𝑎1𝑎𝑠,𝑎𝑡;𝑞_∞=𝑏𝑐𝑡;𝑞_∞𝑎𝑠,𝑎𝑡,𝑐𝑡;𝑞_∞²Φ₁.𝑏,𝑎𝑡𝑏𝑐𝑡;𝑞,𝑠𝑐 (35)

Throughout this paper, we also often use the following property.

Lemma 24. If |𝑠𝑐|<1, |𝑡𝑐|<1, then 𝐹𝑏;𝑐𝐷_𝑞𝑎𝑎^𝑀𝑎𝜔;𝑞_∞𝑎𝑠,𝑎𝑡;𝑞_∞=𝑎^𝑀𝑞;𝑞_𝑀𝑎𝜔,𝑏𝑐𝑡;𝑞_∞𝑎𝑠,𝑎𝑡,𝑐𝑡;𝑞_∞×^∞_𝑘=0(𝑐/𝑎)^𝑘𝑏,𝑎𝑠,𝑎𝑡;𝑞_𝑘𝑞,𝑎𝜔,𝑏𝑐𝑡;𝑞_𝑘𝑞;𝑞_𝑀−𝑘×₃Φ₂𝑏𝑞^𝑘,𝑎𝑡𝑞^𝑘,𝜔𝑠𝑎𝜔𝑞^𝑘,𝑏𝑐𝑡𝑞^𝑘;𝑞,𝑠𝑐. (36)

Proof. We find that LHS of Lemma 24 equates to ^∞_𝑛=0𝑏;𝑞_𝑛𝑐^𝑛𝑞;𝑞_𝑛^𝑛_𝑘=0𝑛𝑘𝑞^{𝑘(𝑘−𝑛)}𝐷^𝑘_𝑞𝑎^𝑀×𝐷_𝑞^𝑛−𝑘𝑎𝜔𝑞^𝑘;𝑞_∞𝑎𝑠𝑞^𝑘,𝑎𝑡𝑞^𝑘;𝑞_∞=^∞_𝑘=0𝑞;𝑞_𝑀𝑎^𝑀−𝑘𝑞;𝑞_𝑘𝑞;𝑞_𝑀−𝑘×^∞_𝑛=𝑘𝑏;𝑞_𝑛𝑐^𝑛𝑞^{𝑘(𝑘−𝑛)}𝑞;𝑞_𝑛−𝑘𝐷_𝑞^𝑛−𝑘𝑎𝜔𝑞^𝑘;𝑞_∞𝑎𝑠𝑞^𝑘,𝑎𝑡𝑞^𝑘;𝑞_∞=𝑎^𝑀𝑞;𝑞_𝑀^∞_𝑘=0(𝑐/𝑎)^𝑘𝑏;𝑞_𝑘𝑞;𝑞_𝑘𝑞;𝑞_𝑀−𝑘𝐹𝑏𝑞^𝑘;𝑐𝑞^𝑘𝐷_𝑞×𝑎𝜔𝑞^𝑘;𝑞_∞𝑎𝑠𝑞^𝑘,𝑎𝑡𝑞^𝑘;𝑞_∞=𝑎^𝑀𝑞;𝑞_𝑀𝑎𝜔,𝑏𝑐𝑡;𝑞_∞𝑎𝑠,𝑎𝑡,𝑐𝑡;𝑞_∞×^∞_𝑘=0(𝑐/𝑎)^𝑘𝑏,𝑎𝑠,𝑎𝑡;𝑞_𝑘𝑞,𝑎𝜔,𝑏𝑐𝑡;𝑞_𝑘𝑞;𝑞_𝑀−𝑘×₃Φ₂𝑏𝑞^𝑘,𝑎𝑡𝑞^𝑘,𝜔𝑠𝑎𝜔𝑞^𝑘,𝑏𝑐𝑡𝑞^𝑘;𝑞,𝑠𝑐. (37) This completes the proof.

3. Proof of Theorems

Proof of Theorem 1. LHS of (3) is equal to ^∞_𝑛=0𝐹𝑎;𝑥𝐷_𝑞𝑦𝑦^(𝑚+𝑛)𝐹𝑏;𝑋𝐷_𝑞𝑌𝑌^(𝑛)𝑧^𝑛𝑞;𝑞_𝑛=𝐹𝑎;𝑥𝐷_𝑞𝑦𝐹𝑏;𝑋𝐷_𝑞𝑌𝑦^𝑚𝑌𝑦𝑧;𝑞_∞=𝐹𝑎;𝑥𝐷_𝑞𝑦𝑦^𝑚𝑦𝑏𝑋𝑧;𝑞_∞𝑦𝑌𝑧,𝑦𝑋𝑧;𝑞_∞=𝑦^𝑚𝑞;𝑞_𝑚𝑏𝑦𝑋𝑧,𝑎𝑥𝑋𝑧;𝑞_∞𝑦𝑌𝑧,𝑦𝑋𝑧,𝑥𝑋𝑧;𝑞_∞×^∞_𝑘=0𝑥/𝑦^𝑘𝑎,𝑦𝑌𝑧,𝑦𝑋𝑧;𝑞_𝑘𝑞,𝑏𝑦𝑋𝑧,𝑎𝑥𝑋𝑧;𝑞_𝑘𝑞;𝑞_𝑚−𝑘×₃Φ₂𝑎𝑞^𝑘,𝑦𝑋𝑧𝑞^𝑘,𝑏𝑋𝑌𝑏𝑦𝑋𝑧𝑞^𝑘,𝑎𝑥𝑋𝑧𝑞^𝑘;𝑞,𝑥𝑌𝑧. (38) We complete the proof.

Proof of Theorem 2. LHS of (4) is equal to 𝐹𝑎;𝑥𝐷_𝑞𝑦1𝑦𝑡∶𝑞_∞𝐹𝑏;𝑋𝐷_𝑞𝑌1𝑌𝑦𝛾,𝑌𝑧;𝑞_∞=𝑏𝑋𝑧;𝑞_∞𝑌𝑧,𝑋𝑧;𝑞_∞^∞_𝑚=0𝑏,𝑌𝑧;𝑞_𝑚𝑋𝛾^𝑚𝑞;𝑏𝑋𝑧_𝑚×𝐹𝑎;𝑥𝐷_𝑞𝑦𝑦^𝑚𝑦𝑡,𝑦𝑌𝛾∶𝑞_∞; (39) then using Lemma 24 and simplifying, we can get the desired result.

Proof of Theorem 3. LHS of (5) is equal to 𝐹𝑎;𝑥𝐷_𝑞𝑦𝑦^𝑛𝐹𝑏;𝑋𝐷_𝑞𝑌𝑌^𝑚𝑌𝑦𝑧;𝑞_∞=𝑌^𝑚𝑞;𝑞_𝑚^∞_𝑘=0(𝑋/𝑌)^𝑘𝑏;𝑞_𝑘𝑞;𝑞_𝑘𝑞;𝑞_𝑚−𝑘𝐹𝑎;𝑥𝐷_𝑞𝑦×𝑦^𝑛𝑦𝑏𝑋𝑧𝑞^𝑘;𝑞_∞𝑦𝑌𝑧𝑞^𝑘,𝑦𝑋𝑧;𝑞_∞=𝑦^𝑛𝑌^𝑚𝑞;𝑞_𝑛𝑞;𝑞_𝑚𝑦𝑏𝑋𝑧,𝑥𝑎𝑋𝑧;𝑞_∞𝑦𝑌𝑧,𝑦𝑋𝑧,𝑥𝑋𝑧;𝑞_∞×^∞_𝑘=0(𝑋/𝑌)^𝑘𝑏,𝑦𝑌𝑧;𝑞_𝑘𝑞,𝑦𝑏𝑋𝑧;𝑞_𝑘𝑞;𝑞_𝑚−𝑘×^∞_𝑗=0𝑥/𝑦^𝑗𝑎,𝑦𝑌𝑧𝑞^𝑘,𝑦𝑋𝑧;𝑞_𝑗𝑞,𝑦𝑏𝑋𝑧𝑞^𝑘,𝑥𝑎𝑋𝑧;𝑞_𝑗𝑞;𝑞_𝑛−𝑗×₃Φ₂𝑎𝑞^𝑗,𝑦𝑋𝑧𝑞^𝑗,𝑏𝑋𝑌𝑏𝑦𝑋𝑧𝑞^𝑘+𝑗,𝑎𝑥𝑋𝑧𝑞^𝑗;𝑞,𝑥𝑌𝑧𝑞^𝑘. (40) This completes the proof.

Proof of Theorem 4. LHS of (6) is equal to ^∞_{𝑚1,𝑚2,𝑚3=0}𝐹𝑎₁;𝑥₁𝐷_𝑞𝑦1𝑦^{𝑚1+𝑚2+𝑚3}₁𝐹𝑎₂;𝑥₂𝐷_𝑞𝑦2×𝑦^𝑚1+𝑚3₂𝐹𝑎₃;𝑥₃𝐷_𝑞𝑦3𝑦^𝑚1+𝑚2₃×𝑡^𝑚1₁𝑡^𝑚2₂𝑡^𝑚3₃𝑞;𝑞_𝑚1𝑞;𝑞_𝑚2𝑞;𝑞_𝑚3=𝐹𝑎₁;𝑥₁𝐷_𝑞𝑦1𝐹𝑎₂;𝑥₂𝐷_𝑞𝑦2𝐹𝑎₃;𝑥₃𝐷_𝑞𝑦3×1𝑡₁𝑦₁𝑦₂𝑦₃,𝑡₂𝑦₁𝑦₃,𝑡₃𝑦₁𝑦₂;𝑞_∞=𝐹𝑎₁;𝑥₁𝐷_𝑞𝑦1𝐹𝑎₂;𝑥₂𝐷_𝑞𝑦2×1𝑡₃𝑦₁𝑦₂;𝑞_∞𝐹𝑎₃;𝑥₃𝐷_𝑞𝑦3×1𝑡₁𝑦₁𝑦₂𝑦₃,𝑡₂𝑦₁𝑦₃;𝑞_∞=𝐹𝑎₁;𝑥₁𝐷_𝑞𝑦1𝐹𝑎₂;𝑥₂𝐷_𝑞𝑦2×𝑎₃𝑥₃𝑡₂𝑦₁;𝑞_∞𝑡₂𝑦₁𝑦₃,𝑥₃𝑡₂𝑦₁;𝑞_∞^∞_𝑗1=0𝑎₃,𝑡₂𝑦₁𝑦₃;𝑞_𝑗1𝑞,𝑎₃𝑥₃𝑡₂𝑦₁;𝑞_𝑗1×𝑥₃𝑡₁𝑦₁𝑦₂^𝑗1𝑡₃𝑦₁𝑦₂,𝑡₁𝑦₁𝑦₂𝑦₃;𝑞_∞=𝐹𝑎₁;𝑥₁𝐷_𝑞𝑦1×𝑎₃𝑥₃𝑡₂𝑦₁;𝑞_∞𝑡₂𝑦₁𝑦₃,𝑥₃𝑡₂𝑦₁;𝑞_∞^∞_𝑗1=0𝑎₃,𝑡₂𝑦₁𝑦₃;𝑞_𝑗1𝑞,𝑎₃𝑥₃𝑡₂𝑦₁;𝑞_𝑗1×𝐹𝑎₂;𝑥₂𝐷_𝑞𝑦2𝑥₃𝑡₁𝑦₁𝑦₂^𝑗1𝑡₃𝑦₁𝑦₂,𝑡₁𝑦₁𝑦₂𝑦₃;𝑞_∞=^∞_𝑗1=0𝑎₃;𝑞_𝑗1𝑥₃𝑡₁𝑦₂^𝑗1𝑞;𝑞_𝑗1𝑞;𝑞_𝑗1×^∞_𝑗2=0𝑥₂/𝑦₂^𝑗2𝑎₂;𝑞_𝑗2𝑞;𝑞_𝑗2𝑞;𝑞_{𝑗1−𝑗2}×^∞_𝑗3=0𝑥₂𝑡₃^𝑗3𝑎₂𝑞^𝑗2;𝑞_𝑗3𝑞;𝑞_𝑗3𝐹𝑎₁;𝑥₁𝐷_𝑞𝑦1×𝑦^𝑗1+𝑗3₁𝑎₃𝑥₃𝑡₂𝑦₁𝑞^𝑗1,𝑎₂𝑥₂𝑡₁𝑦₁𝑦₃𝑞^𝑗2+𝑗3;𝑞_∞×𝑡₂𝑦₁𝑦₃𝑞^𝑗1,𝑡₃𝑦₁𝑦₂𝑞^𝑗2,𝑡₁𝑦₁𝑦₂𝑦₃𝑞^𝑗2+𝑗3,𝑥₃𝑡₂𝑦₁,𝑥₂𝑡₁𝑦₁𝑦₃;𝑞_∞⁻¹. (41)
Setting 𝑓𝑦₁=𝑦^𝑗1+𝑗3₁𝑎₃𝑥₃𝑡₂𝑦₁𝑞^𝑗1,𝑎₂𝑥₂𝑡₁𝑦₁𝑦₃𝑞^𝑗2+𝑗3;𝑞_∞×𝑡₂𝑦₁𝑦₃𝑞^𝑗1,𝑡₃𝑦₁𝑦₂𝑞^𝑗2,𝑡₁𝑦₁𝑦₂𝑦₃𝑞^𝑗2+𝑗3,𝑥₃𝑡₂𝑦₁,𝑥₂𝑡₁𝑦₁𝑦₃;𝑞_∞⁻¹, (42) then by applying Lemma 18 and simplifying, we get 𝐹𝑎₁;𝑥₁𝐷_𝑞𝑦1𝑓𝑦₁=𝑦^𝑗1+𝑗3₁𝑎₁𝑥₁𝑦₁,𝑎₃𝑥₃𝑡₂𝑦₁,𝑎₂𝑥₂𝑡₁𝑦₁𝑦₃;𝑞_∞×𝑥₁𝑦₁,𝑡₂𝑦₁𝑦₃,𝑡₃𝑦₁𝑦₂,𝑡₁𝑦₁𝑦₂𝑦₃,𝑥₃𝑡₂𝑦₁,𝑥₂𝑡₁𝑦₁𝑦₃;𝑞_∞⁻¹×^∞_𝑗4=0𝑎₁,𝑥₃𝑡₂𝑦₁,𝑥₂𝑡₁𝑦₁𝑦₃;𝑞_𝑗4(−1)^𝑗4𝑞^{−(𝑗4−1)𝑗4/2}𝑞,𝑥₁/𝑦₁𝑞^𝑗4;𝑞_𝑗4×𝑥₁𝑞^𝑗1+𝑗3𝑦₁^𝑗4𝑡₃𝑦₁𝑦₂;𝑞_𝑗2+𝑗4𝑡₂𝑦₁𝑦₃;𝑞_𝑗1+𝑗4𝑎₃𝑥₃𝑡₂𝑦₁;𝑞_𝑗1+𝑗4×𝑡₁𝑦₁𝑦₂𝑦₃;𝑞_{𝑗2+𝑗3+𝑗4}𝑎₂𝑥₂𝑡₁𝑦₁𝑦₃;𝑞_{𝑗2+𝑗3+𝑗4}; (43) substituting it into (41), we complete the theorem.

Proof of Theorem 5. LHS of (7) is equal to 𝐹𝑎₁;𝑥₁𝐷_𝑞𝑦1𝐹𝑎₂;𝑥₂𝐷_𝑞𝑦2𝐹𝑎₃;𝑥₃𝐷_𝑞𝑦3×1𝑦₁𝑦₂𝑡₁,𝑦₁𝑦₂𝑡₂,𝑦₁𝑦₃𝑡₃;𝑞_∞=𝐹𝑎₁;𝑥₁𝐷_𝑞𝑦1𝐹𝑎₂;𝑥₂𝐷_𝑞𝑦2×𝑎₃𝑥₃𝑦₁𝑡₃;𝑞_∞𝑦₁𝑦₃𝑡₃,𝑦₁𝑥₃𝑡₃;𝑞_∞1𝑦₁𝑦₂𝑡₁,𝑦₁𝑦₂𝑡₂;𝑞_∞=^∞_𝑗1=0𝑎₂;𝑞_𝑗1𝑡₁𝑥₂^𝑗1𝑞;𝑞_𝑗1𝐹𝑎₁;𝑥₁𝐷_𝑞𝑦1×𝑦^𝑗1₁𝑎₃𝑦₁𝑥₃𝑡₃,𝑎₂𝑦₁𝑥₂𝑡₂𝑞^𝑗1;𝑞_∞×𝑦₁𝑦₃𝑡₃,𝑦₁𝑥₃𝑡₃,𝑦₁𝑦₂𝑡₁,𝑦₁𝑦₂𝑡₂𝑞^𝑗1,𝑦₁𝑥₂𝑡₂;𝑞_∞⁻¹. (44)
Applying Lemma 18 and simplifying, we get the theorem. This completes the proof.

Proof of Theorems 6 and 7. Using Lemma 17, then Theorem 6 can be obtained by Lemma 20 directly.
Using Lemma 17 and then applying Lemma 18, we can get Theorem 7.

Proof of Theorem 8. LHS of (10) is equal to 𝐹𝑎;𝑥𝐷_𝑞𝑦𝑦𝑡₁,𝑦𝑡₄;𝑞_∞𝑦𝑡₂,𝑦𝑡₃;𝑞_∞=^∞_𝑛=0𝑎;𝑞_𝑛𝑥^𝑛𝑞;𝑞_𝑛𝐷^𝑛_𝑞𝑦𝑡₁,𝑦𝑡₄;𝑞_∞𝑦𝑡₂,𝑦𝑡₃;𝑞_∞=^∞_𝑛=0𝑎;𝑞_𝑛𝑥^𝑛𝑞;𝑞_𝑛×^𝑛_𝑘=0𝑛𝑘𝑞^{𝑘(𝑘−𝑛)}𝐷^𝑘_𝑞𝑦𝑡₁;𝑞_∞𝑦𝑡₂;𝑞_∞×𝐷_𝑞^𝑛−𝑘𝑦𝑡₄𝑞^𝑘;𝑞_∞𝑦𝑡₃𝑞^𝑘;𝑞_∞=𝑦𝑡₁,𝑦𝑡₄;𝑞_∞𝑦𝑡₂,𝑦𝑡₃;𝑞_∞^∞_𝑛=0(𝑎;𝑎)_𝑛𝑥𝑡₃^𝑛𝑞;𝑞_𝑛×^𝑛_𝑘=0𝑞;𝑞_𝑛𝑡₂/𝑡₃^𝑘𝑞;𝑞_𝑘𝑞;𝑞_𝑛−𝑘𝑡₁/𝑡₂,𝑦𝑡₃;𝑞_𝑘𝑡₄/𝑡₃;𝑞_𝑛−𝑘𝑦𝑡_𝑞;𝑞_𝑘=𝑦𝑡₁,𝑦𝑡₄;𝑞_∞𝑦𝑡₂,𝑦𝑡₃;𝑞_∞^∞_𝑗=0𝑎,𝑡₁/𝑡₂,𝑦𝑡₃;𝑞_𝑗𝑥𝑡₂^𝑗𝑞,𝑦𝑡₁,𝑦𝑡₄;𝑞_𝑗×₂Φ₁𝑎𝑞^𝑗,𝑡₄𝑡₃𝑦𝑡₄𝑞^𝑗;𝑞,𝑥𝑡₃; (45) we complete the proof.

Proof of Theorem 9. LHS of (11) equates to 𝐹𝑎;𝑥𝐷_𝑞𝑦𝐹𝑎₀;𝑥₀𝐷_𝑞𝑦0𝐹𝑎₁;𝑥₁𝐷_𝑞𝑦1×𝐹𝑎₂;𝑥₂𝐷_𝑞𝑦2𝐹𝑎₃;𝑥₃𝐷_𝑞𝑦3×𝑦^𝑠𝑦^𝑟₀𝑦𝑦₀𝑡,𝑦𝑦₁𝑡₁,𝑦𝑦₂𝑡₂,𝑦𝑦₃𝑡₃;𝑞_∞=𝐹𝑎;𝑥𝐷_𝑞𝑦𝐹𝑎₀;𝑥₀𝐷_𝑞𝑦0𝐹𝑎₁;𝑥₁𝐷_𝑞𝑦1×𝐹𝑎₂;𝑥₂𝐷_𝑞𝑦2𝑦^𝑠𝑦^𝑟₀𝑦𝑦₀𝑡,𝑦𝑦₁𝑡₁,𝑦𝑦₂𝑡₂;𝑞_∞×𝑎₃𝑦𝑥₃𝑡₃;𝑞_∞𝑦𝑦₃𝑡₃,𝑦𝑥₃𝑡₃;𝑞_∞=𝐹𝑎;𝑥𝐷_𝑞𝑦𝐹𝑎₀;𝑥₀𝐷_𝑞𝑦0𝐹𝑎₁;𝑥₁𝐷_𝑞𝑦1×𝑦^𝑠𝑦^𝑟₀𝑦𝑦₀𝑡,𝑦𝑦₁𝑡₁;𝑞_∞𝑎₂𝑦𝑥₂𝑡₂;𝑞_∞𝑦𝑦₂𝑡₂,𝑦𝑥₂𝑡₂;𝑞_∞×𝑎₃𝑦𝑥₃𝑡₃;𝑞_∞𝑦𝑦₃𝑡₃,𝑦𝑥₃𝑡₃;𝑞_∞=𝐹𝑎;𝑥𝐷_𝑞𝑦𝐹𝑎₀;𝑥₀𝐷_𝑞𝑦0𝑦^𝑠𝑦^𝑟₀𝑦𝑦₀𝑡;𝑞_∞×𝑎₁𝑦𝑥₁𝑡₁;𝑞_∞𝑦𝑦₁𝑡₁,𝑦𝑥₁𝑡₁;𝑞_∞𝑎₂𝑦𝑥₂𝑡₂;𝑞_∞𝑦𝑦₂𝑡₂,𝑦𝑥₂𝑡₂;𝑞_∞×𝑎₃𝑦𝑥₃𝑡₃;𝑞_∞𝑦𝑦₃𝑡₃,𝑦𝑥₃𝑡₃;𝑞_∞. (46) If we set 𝑓𝑦=𝐹𝑎₀;𝑥₀𝐷_𝑞𝑦0𝑦^𝑠𝑦^𝑟₀𝑦𝑦₀𝑡;𝑞_∞×𝑎₁𝑦𝑥₁𝑡₁;𝑞_∞𝑦𝑦₁𝑡₁,𝑦𝑥₁𝑡₁;𝑞_∞𝑎₂𝑦𝑥₂𝑡₂;𝑞_∞𝑦𝑦₂𝑡₂,𝑦𝑥₂𝑡₂;𝑞_∞×𝑎₃𝑦𝑥₃𝑡₃;𝑞_∞𝑦𝑦₃𝑡₃,𝑦𝑥₃𝑡₃;𝑞_∞, (47) using Lemma 24, we get 𝑓(𝑦); then employing Lemma 18, we can get the desired result.

Proof of Theorem 10. LHS of (12) equates to 𝐹𝑎₁;𝑥₁𝐷_𝑞𝑦1𝐹𝑎₂;𝑥₂𝐷_𝑞𝑦2𝐹𝑎₃;𝑥₃𝐷_𝑞𝑦3×𝐹𝑎₄;𝑥₄𝐷_𝑞𝑦4𝑡₁𝑦₂𝑦₃𝑦₄,𝑡₄𝑦₁𝑦₂𝑦₃;𝑞_∞𝑡₂𝑦₁𝑦₃𝑦₄,𝑡₃𝑦₁𝑦₂𝑦₄;𝑞_∞; (48) then using Lemma 18 four times we can get the result.

4. Some Other Cases and a Homogeneous 𝑞-Mehler’s Formula

In this section we obtain Mehler’s formula involving a ₃Φ₂ series and then give an interesting relationship between the polynomial multiplications.

Theorem 25 (cf. [1, Equation (1.17)] or [7, Lemma 12]). One has ^∞_𝑛=0𝜙